This is going to end up reiteratig a lot of advice, and disagreeing with some. That advice it does disagree with, is not so much wrong as pointed towards doing something other than what you're trying to do.
What with the flattening, and the three stage breading, we can safely assume you're trying to cook your chicken "schnitzel" style (not that it doesn't have a variety of other names). In any case, the breading technique is "flour, egg, crumb" which is substantially the same as it's kissing cousin, "flour, egg, flour." The underlying pan technique is "pan frying."
Pan frying is not searing or sauteeing, and unfortunately, the distinction hasn't always been observed through this thread. It started to get confusing, and that's largely why I decided to jump in.
There are nine most likely reasons the crust might be as fragile, and as prone to come off as you describe.
1. You're chicken is too wet. When people talk about a flour - egg - flour or flour - egg - crumb breading they talk about how the first flouring is meant to prepare the surface for the second; but not what it's actually doing.
Ideally, the flour absorbs excess surface mixture so the egg wash is the only moisture under the final breading. However, if there's too much moisture on the surface of the chicken breast (or thigh), it will overwhelm the flour.
So, before and/or after flattening, dry the chicken with a paper towel.
2. Too much or too little first-coat flour. Don't dust, dredge; don't dredge heavily; and shake off any excess flour before sending the flour to the egg.
3. Your egg wash is too dilute. This isn't a common problem, and you can actually use a fairly thin wash; but just in case, we might as well go through it. If you're diluting your egg wash (and most people don't) with milk, juice or some other liquid flavoring, don't use too much. Water, is unnecessary unless you only have one egg in the house, no other liquid, and you're trying to make that egg work like two.
Hot sauce is a wonderful thing. But presumably your sense of self-preservation will limit your use long before over-dilution becomes a problem.
DO season your egg wash.
4. You're dredging wrong and the breading is not adhering to the chicken as a result.
Don't use tongs or a fork to dredge. Use the "dry hand, wet hand" method, instead. Tongs work for some people, but it's tricky. Forget them for now.
(a) Use your "dry hand" (left or right) to take the chicken to the flour, and from the flour to the egg. Don't let it get wet when you put the chicken into the egg.
(b) Use your wet hand (the other one) to turn the chicken in the egg, making sure it's well coated on both sides. Use your wet hand to lift your chicken from the egg wasy to the bread crumbs.
(c) Continue using your wet hand to turn the chicken in the crumbs, making sure it's coated on both sides. If you need to lift some crumbs over the top of the chicken piece, use your dry hand.
(d) Use your dry hand to lift the chicken to a rack. If you don't have a tightly spaced rack, you may have to use a baking sheet.
Note: If you use your wet hand to lift the chicken into the flour to begin with, the flour will not adhere to the wet spot. If you use your wet hand to lift the chicken from the breading, you will strip the breading where your wet fingers touch it.
5. There's excess breading and/or you're not letting your chicken rest after breading and before cooking. Both of these result in a too heavy, too fragile breading that wants to pull away from the chicken as and after it's cooked.
After you've got the chicken on the rack, wash your hands and dry them thoroughly. Use one of your clean, dry hands to pick up the chicken and shake the excess off. Replace it on the rack and let it rest. If you don't have a rack, you'll have to use two pans.
Allow the chicken to rest before cooking for at least seven minutes after breading. If you like, you can let it rest substantially longer. If you want to or must go for more than 30 minutes, put the chicken in a clean pan, cover it loosely with foil or cling, and hold it in the refrigerator. You may hold as long as overnight.
6. Your not properly heating the pan and/or oil. Already well covered, but what the heck.
First preheat the pan over medium-high heat. When it's hot, usually a matter of two or three minutes, add the oil. Wait until the oil is hot before adding the chicken.
The "shimmer" test works very well. If you're not confident with that, tear a little piece off a slice of fread and put it in the oil as a "tester." If it sizzles and browns quickly, voila! If it does not, you need to either raise the flame, or give it some more time -- probably the second. If the bread smokes and blackens, the oil is too hot and the fire should be reduced.
If the oil becomes too hot and smokes heavily, it is ruined and should be replaced. Not only that, the overheated oil will stick tenaciously to the pan, which should be cleaned before reusing. Overheated oil not only tastes bad, it's quite unhealthy. Just get rid of it.
7. You're using too much or too little oil to pan fry.
If you were sauteeing or searing you'd add very little oil ala Chef Todd's post, but you're not.
You're pan frying. The exact right amount of oil is enough to come almost but not quite half way up the side of the food to be fried. The exact right amount of oil is enough to come almost but not quite half way up the side of the food to be fried.
Nobody gets the "exact right amount of oil" everytime. Do your best and remember, better a little too much than too little. If you're a little short, you can add a little more during the process, but better to add between batches.
8. You're overcrowding the pan. Don't. In addition to playing hob with temperature control. The uncooked breading from one piece will break the adhesion from the breading on another. And vice versa.
9. You're handling the chicken too roughly in the pan, and as you lift it out. Use an appropriate spatula or tongs.
The half melted, plastic spatula that came with your George Foremean grill is not appropriate. Get a good, metal spatula -- preferably a "fish turner" -- everyone should have at least one good spatula.
Pros don't have any trouble using spaghetti tongs for anything, but when you use them they probably rip up everything they touch. They're cheap, but they are not your friend. Seek out something gentler, like wood tongs, tongs with silicone coated tips or, the Westmark "spatula" tongs. The Westmarks are awesome.
Some other thoughts:
You may absolutely use panko. A panko coating is a little more delicate than a dried or stale crumb coating, but so what? The Japanese use panko for making exactly what you're making and so do I. Why not you?
Season everything: Season the chicken, the flour, the egg, and (if possible) the crumbs. Slightly underseason the chicken. Use enough salt and pepper and paprika (if using) so that the flour appears "dirty." Undersalt the egg, but by all means use hot sauce. Breadcrumbs don't season particularly well in the nominal sense of "seasoning;" but do add some fresh herbs and perhaps a bit of dry, grated cheese (cotija, parmesan, etc.) too. Keep your eye on the totality of salt so as not to oversalt, but remember that underseasoning is a far more common fault than overdoing it. Waiting to salt until the food gets to the table makes for a poor dish. On the other hand, seasoning in layers makes even simple seasoning taste "right."
Hope this helps,