Sounds like you're getting a natural jus which is an artifact of low temperature, tightly covered cooking. There are a lot of ways to make it happen. More about that in a minute.
Apparently anything cooked "low and slow," in your neck of the country is called "barbecue." Other parts have different names. But since you live where you live, and are asking more about food than words, we won't worry too much about it if you don't want to.
Probably an artifact of not hurrying the cooking, and whatever sort of seasoning is used. From your description it sounds very simple.
Worth a shot. If the number one place won't give you their secret, maybe one of the other's will get you close. If not, it's no biggie. I'm sure we can approach the texture, juiciness and taste you like from the sliced pork you're getting.
The first place to start is the cut. Ask around before you go shopping. Since you're eating it sliced instead of pulled it's probably, butt, picnic, whole shoulder or loin. The easiest and most forgiving cut is the butt. For sliced pork, I'd suggest starting with half a butt.
If it's boneless, you'll want to truss it so it holds together during the cook. If it has a lot of surface fat -- don't trim it closer than 1/4". Try and leave some. Most of it will render off.
The first step is to preheat your oven. 225 is a sort of magic temperature for pork barbecue, but it will take quite some time. There are some safety issues cooking that slowly indoors too. For indoor cooking, 275 is very good and takes significantly less time. So, preheat to 275.
Next, comes the "slather." Slathering means to rub the pork with something that will help the dry seasonings stick to the outside of the roast. Competition barbecue is mostly southern and frequently starts with a "mustard slather," surprisingly simple yellow mustard leaves very little color or taste. Some people use oil, some use a paste of pureed onion. I find mayonnaise works very well, plus it helps me get a little extra flavor in if I mix it with thinner sauces. So, why not mix some Worcestershire sauce with the mayo, add a few splashes of hot sauce, and rub the surface with just enough to get it greased.
Then rub the meat with a dry seasoning rub made similar to this one:
1/2 cup brown sugar
3 tbs paprika
3 tbs kosher salt (or 1-1/2 tbs table salt, but kosher sticks to the meat better)
1-1/2 tbs freshly ground, coarse black pepper
1 tbs granulated onion (better than powdered onion)
1 tbs granulated garlic (better than powdered garlic)
1 tsp celery salt
1 tsp poultry seasoning
Season generously, there's a lot of meat in that roast and not much surface area. Since you're not going to be adding seasoning later, this is your last chance to get any seasoning on the meat.
Make 2 cups of mock pork stock by mixing 1 cup each beef and chicken stock. You can add a couple ounces of apple juice, white grape juice, bourbon, medium dry white wine (like a dry Riesling, Gewurztraminer, or Liebfraumilch) sherry or port to the mix if you like. Guess which ones I like. Guess which ones leave me cold.
Rough chop an onion, a stalk of celery and a carrot.
Mound your vegetables on the bottom of the pan, so they form a flat bed for the roast. Put the roast on top of them. Add the stock to the bottom of the pan. Cover the pan with foil, as tightly as possible. If you're using a large, heavy casserole or rondeau, with a cover, so much the better. Put the meat in the preheated oven. A four pound boneless roast will take about four hours to get to an internal temperature of 180* to 190*F. Anything less won't be tender. Anything more will fall apart like pulled pork. I suggest allowing yourself plenty of slack on the cook-time. Start measuring the meat's internal temperature at the 45 minutes per pound stage (3 hours for a 4 pound roast). Sometimes you'll experience a "stall" when cooking meat at a low temperature. That means that the meat's internal temperature seems to stop going up -- sometimes for hours. However, with a cut of meat that small, and a temperature as high as 275, it's unlikely you'll experience much stall a-tall.
When the meat is cooked, pour off the drippings and juices and reserve. Wrap the meat in aluminum foil and allow to rest for at least half an hour. You can let it rest longer in the oven with the fire off, or longer still in an insulated cooler. Up to four hours or so, the longer the better. (A good way of planning is to estimate cooking time and allow 2-1/2 to 3 hours of rest time. That way you're covered if the cook goes slow or fast. Fortunately, 275* will give you a fairly predictable cook, at least compared to 225*.)
Meanwhile strain off most, but not all, of the fat. (Pork fat is very palatable, free from transfat, and will carry a lot of pork and seasoning flavor to the gravy.) Reheat the gravy. Taste and adjust for salt and pepper. Hold the gravy hot, at a bare simmer. If the "sauce" in your part of the country is at all thick, you may want to thicken the sauce slightly by pureeing the vegetables with the pan juices, pressing the vegetables into the juices through a sieve (recommended), and/or with a cornstarch slurry. Without thickening it's a jus, slightly thickened it's a jus lie. Quiz tomorrow.
Unwrap the pork and place it on a carving board with a "gravy channel" or other "catcher." If you cooked a boneless, tied roast you can use a long thin knife to cut very thin, even slices. If you cooked a bone-in shoulder, you'll have to take off large chunks the best way you can, then slice them. In any case, you want thin slices. "Thin slices" is another way of saying "a dull knife is not your friend." Catch whatever juices run from the meat and add them to the au jus gravy.
Split a roll, ladle a little of the jus on the bottom, then mound some pork on it and add a little jus to the meat. Finally, pour just enough jus to the top to moisten it. Set the top to the side of the plate and serve open-faced along with whatever garniture (lettus, onions, tomato, cole-slaw, etc.) is customary in your neck of the woods.
That oughtta 'bout do 'er,