How to Make It
For the cheeseburgers: Grind the chuck and flank steak through the large die of the meat grinder into a bowl. Mix gently to combine. Grind half the mixture through the small die. Mix the two halves together.
Portion the meat into ten 3-ounce balls; each burger will get 2 patties. (The balls can be shaped ahead, tightly covered, and refrigerated for up to 1 day. Remove from the refrigerator about 30 minutes before you’re ready to cook; it’s important that the balls are not ice-cold when they hit the hot pan.)
Generously butter the cut sides of the buns. Toast on a griddle until golden brown. Transfer to a rimmed baking sheet, toasted side up, and set aside.
Heat two 12-inch cast-iron skillets over high heat until very hot and just barely beginning to smoke, 2 to 3 minutes. Divide the balls between the two hot skillets, smashing each one down with a spatula. When the patties are charred on the bottom, with a dark brown crust, about 2 minutes, turn them over. Place the onion slices on top of 5 of the patties and place a slice of cheese on each patty. Cook for another 1 to 2 minutes, until the onions have softened and the cheese has melted; the burgers will be cooked to medium. Stack the patties without onions on top of the patties with onions. Remove from the heat.
Smear both cut sides of the buns with the sauce. Place 5 pickles on the bottom of each bun. Add the burgers and cover with the tops of the buns.
For the sauce: Combine all the ingredients for the sauce in a large container and stir to blend well. Cover and refrigerate. (Tightly covered, the sauce will keep for up to 5 days in the refrigerator.)
I always grind the meat for my burgers myself, using a mix of chuck and flank steak. The freshness really comes through. But if you don’t have a meat grinder, you can have the butcher do this—buy the meat the day you plan to make the burgers.