Prepare chicken: Heat a grill to medium (350° to 450°) with an area left clear of charcoal or a gas burner turned off to make an indirect heat area. In a medium bowl, combine vinegar, achiote, oil, and salt; add chicken and turn to coat well. Grill over indirect heat with lid down, turning once, until chicken is cooked through, about 20 minutes. Let cool, then shred meat from bones. Set aside.
Meanwhile, make panuchos: Purée beans and their liquid in a food processor, adding water if needed so they're very creamy. In a medium frying pan, cook onion in oil over medium heat until golden, 5 minutes. Stir in beans and cook until bubbling, 1 to 2 minutes. Set aside, covered.
Combine masa and salt in a bowl, adding a bit of water (or flour) if needed so masa is pliable but not sticky. Divide into 15 equal balls, set on a baking sheet, and cover with plastic wrap.
Heat a heavy medium griddle or skillet, preferably cast iron, over medium heat until hot. Working with 1 masa ball at a time, line a tortilla press with a heavy piece of plastic (a freezer bag cut in half works well). Center masa ball on top and cover with another piece of plastic. Press tortilla to 4 to 5 in. wide.
Peel plastic from tortilla (if tortilla sticks, dust with a little flour and reroll) and put tortilla on hot griddle. When underside no longer sticks and is lightly speckled, about 1 minute, flip and cook 1 minute on other side. Flip again and with a folded clean kitchen towel, firmly press tortilla all over (this helps it puff). Cook until it puffs and the underside is light golden, 15 to 30 more seconds. Transfer tortilla to a work surface.
Slit hot tortilla carefully 1/2 in. from edge to make an opening about 2 in. long. Without tearing tortilla, slide a small metal spatula inside to create a pocket. Let cool. Repeat steps 4, 5, and 6 for each tortilla.
Open up the pocket of a cooled tortilla with your finger, squeeze gently to hold it open, and smear a rounded spoonful of beans inside. Pat top of tortilla with your fingers to spread beans. Fill each panucho the same way.
Make toppings: In a bowl, combine 1/4 cup water, the vinegar, salt, and onion. Chill until used. Thinly slice lettuce and cut 15 thin slices each of tomato and avocado.
Preheat oven to 250° and set a rimmed baking sheet inside. Heat a large skillet, preferably cast iron, over medium heat. Add 2 tbsp. oil and a single layer of panuchos. Cook until golden, turning once, 3 to 5 minutes total. Transfer panuchos to pan in oven. Repeat to cook remaining panuchos, adding more oil as needed.
Drain onion and assemble panuchos: Set them on a platter and top with lettuce, chicken, onion, tomato, and avocado. Serve sauce on the side.
*Find appealingly musky, reddish achiote paste, ground from annatto seeds, at Latino grocery stores; you'll also find prepared masa there. Or, instead of prepared masa, substitute 2 cups masa harina combined with 1 1/3 cups water.