Make burgers: Freeze rib-eye and short ribs until firm but not frozen, 1 to 3 hours.
In a nonstick frying pan over medium heat, cook yellow onion in oil until starting to brown, about 10 minutes. Reduce heat to medium-low; cook, stirring occasionally, until mostly browned, about 15 minutes. Let cool, then chop.
Stack bacon slices. Cut in half lengthwise; then cut crosswise into 1/2-in.-thick pieces. Fry over low heat until crisp, 15 to 20 minutes. Drain.
Cut rib-eye and short-rib meat into 1/2-in. cubes, discarding any membrane or gristle but keeping all the fat so burgers are juicy. In two batches, pulse in a food processor just until coarsely ground and starting to come together. Turn into a bowl and add cooked bacon, caramelized onion, garlic, salt, pepper, chives, chervil, and duck fat. Toss together gently with your fingers spread wide.
Form mix into 4 balls. Toss each ball rapidly between your hands for at least a minute until round and very smooth (see "How to Shape a Burger," below). Smoosh each into a 3/4-in.-thick patty, then pat and push edges smooth with cupped hands; cracks may cause burgers to fall apart when grilled. Chill, covered, at least 2 hours.
Meanwhile, blend mayonnaise with lemon zest, pepper, and mustard; chill until using.
Ignite a chimney of charcoal briquets on firegrate of a charcoal grill. When briquets are covered with ash, 20 minutes, pour onto firegrate. Arrange briquets in a ring big enough to fit a 12-in. cast-iron skillet in center, then move 10 to 12 of them to center of ring and space out evenly. Add 1 qt. small fruitwood chunks (about 1 in. thick) to coal ring. When chunks light, cover grill (open lid vents) and let smoke build for 2 to 3 minutes.
Meanwhile, brush buns all over with butter, then cut in half. Season tomatoes lightly with salt and toss lettuce with vinaigrette. Bring buns, patties, and cheddar (if using) out to the grill.
Put patties in skillet, set on coals in center (within ring), and cook until well browned underneath, 4 to 5 minutes. With a long-handled spatula, carefully turn and cook until browned on second side, 4 to 5 minutes for medium.
As soon as burgers are turned, set cooking grate in place and arrange buns directly over ring of coals, cut side down. Toast until nicely browned and crisp, 4 to 5 minutes.
Using a heatproof glove and potholders, lift skillet to a heatproof spot and transfer burgers to a platter; add cheddar if using, so it melts.
Spread cut sides of buns thinly with seasoned mayonnaise. Make burgers with patties, Roquefort (if using), a slice each of red and yellow tomato, the red onion, chiles, and lettuce.
How to Shape a Burger (aka play burger baseball): "To shape our burgers, our cooks form a team and throw them from hand to hand with a thwack, like we're playing baseball, until they're smooth," says von Blöm. "You're not overworking the inside, but you're going to get a great crust."
*Find duck fat at specialty butcher shops. For vinaigrette recipes, see sunset.com.
Make ahead: Burger patties and seasoned mayonnaise, up to 1 day, chilled airtight.