Ohhhhhhh-So-Good Oklahoma Burgers with Goody-Goody Sauce

In a small saucepan with a lid, melt the butter over very low heat. Add the chopped onion and garlic. Cover and cook, stirring occasionally, until the onion is sweet, light golden, and very soft, about 1 hour. (Do not allow the onion to brown.) Add the tomatoes, celery seeds, salt, and pepper. Cover and cook, stirring occasionally, until the tomatoes soften and the sauce thickens, about 3 hours more. Using a blender, purée the sauce. Transfer to squeeze bottles or an airtight container. Refrigerate until cold, about 2 hours. (The sauce can be refrigerated for up to 2 weeks.)

Heat a large cast-iron skillet or griddle over medium-high.

Using a mandoline or the slicing side of a box grater, shave the onions as thinly as possible.

In a bowl, season the beef with salt and pepper; form into 4 equal portions. Roll the meat into balls.

Drizzle the oil in the skillet or over the meat. Place the meat in the skillet, spacing the balls about 3 inches apart; top each with one-quarter of the onions. Using a spatula, press the onions into the meat until flattened into patties. Cook the patties for 3 to 4 minutes, then flip them over. Pressing the patties often with a spatula, cook for 2 minutes. Top each patty with 2 slices of cheese. Cook until the cheese melts, about 2 minutes more.

Build the burgers with the roll bottoms, pickles, lettuce, patties, sauce, and roll tops.