Prepare the spicy sauce by combining all the sauce ingredients in a bowl. Stir to combine. Refrigerate until ready to use.
Divide the beef into 12 equal size balls (85 g each).
Preheat a griddle or a large skillet over medium-high heat to 425°F to 450°F surface temperature (using an instant read thermometer).
While the griddle preheats, toast the buns until lightly charred.
As these burgers cook very quickly, it is best to prepare the buns before you begin cooking. On the bottom of each bun, layer a spoonful of the sauce, lettuce, tomatoes, onions. Set aside.
Lightly brush the griddle with avocado oil (or other high heat tolerant oil of your choice).
Working in batches, place balls of beef on the griddle, cover with parchment paper and smash down with a burger press (or use a stiff metal spatula and a second spatula to add pressure). The smashed patties should be slightly wider than the burger bun. Uneven edges are expected and welcomed.
Season liberally with salt & pepper and cook, undisturbed, until outer edges are brown, about 1.5 to 2 minutes.
Using a stiff metal spatula, carefully scrape patties from the griddle, making sure to keep all of the browned crust on the patty. Flip the patties, season with salt & pepper.
On half of the patties, place about 20g of the shredded smoked cheddar cheese, and on the remaining patties add a slice of processed cheese. Cook until the cheese begins to melt, about 1 minute. Place each patty with processed cheese on top of a patty with the smoked cheddar, then move the double patties to an upper rack to keep warm. Repeat until all burgers are cooked.
To cook the eggs, you can either use the griddle, or cook them in a skillet while the last few patties are cooking on the grill.