Just a couple notes here.
1. Knives
BDL's mildly snide crack about sushi chefs and the like isn't about a distinction between big chef's knives and thin filleting knives. The complex Japanese technique he sketched also depends on a thick, heavy, and totally inflexible knife called a deba-bocho. The idea is that the freakishly sharp, hard, and durable edge combines with the technique to produce a series of cuts that perfectly conform to the shape and structure of the individual fish you're working with. And I will say that the results speak for themselves: nobody can produce that glass-smooth surface with a flexible fillet knife and the French technique. But it takes an awful lot of practice, a good (and rather expensive) knife, and a lot of practice sharpening.
2. Eels
Here I would really advise some approximation of Japanese technique. The knife you want is quite short-bladed, thick, inflexible, and frighteningly sharp. Japanese cooks use eel knives, oddly enough, which come in several regional forms. It would also help a lot to have a glove made of wire wool or the like, so you can hold onto the slimy thing. Anyway, using this knife, follow BDL's directions for a roundfish. Think of it as three cuts (not including removing the head): down the back to the bone, on top of the bone, and down the belly to -- but not through -- the skin. Repeat the cut, laying the fillet flat, cut-side down, to remove the backbone and main skeleton.
Now at this point it's going to depend on the eel. With most eels, just pull out every crosswise bone you can find, using a needle-nosed pliers or the like. Don't expect them to come easily or leave the surface glass-smooth, and expect to spend a good long time at it. But then there are some eels that have zillions of little bones running every which way -- the pike conger eel is the standard example here (known as hamo in Japan). With eels like this, pulling with a pliers will leave you a very small pile of minced meat and no fillet. So before pulling any bones, ask yourself what you're going to do with this fish. If you're going to roast it or something like that, cook it on the bone and remove the bones afterward -- they'll come easily. If you're going to broil it (a la Japanese unagi places), you need to pull all the bones you can, but fortunately broiling it with sauce will cover up the roughness of the fillet surface. Normally you would not eat eel raw -- my impression is that the Japanese consider it unsafe, but I'm not sure.
If on the other hand you're dealing with a pike conger eel (hamo), with a zillion bones everywhere, there are only two solutions that I know of. One is to use it exclusively for making soup and stew -- and you strain all the meat and bone out before serving. The other is to purchase a hamo knife, which is about a foot long, very slightly curved, single-edged, and weighs a ton. Then practice for many months or years: lay the fish skin-side down, horizontally on the board. With your knife, use a steady push-motion beginning slightly above the fish surface and finishing without cutting through the skin, in one smooth cut shearing through all bones. Repeat this cut about 1mm to the left, and then again, and again, and again. Every three inches or so, complete the cut through the skin, so you get roughly 3x3 squares. In the end, you should have sheared every bone into very fine pieces, so that you can eat the flesh with the bones in. Blanch each square fillet briefly, until just barely done, then cool and serve.