Place the chuck, sirloin, and the salt in the bowl of a food processor and pulse until it resembles a coarse, medium grind, about 10 one-second pulses.
Divide the meat into four 4-ounce (113-gram) portions and roll into balls. Place one of the meat balls between two sheets of wax paper on a tortilla press and flatten completely into a 5- to 6-inch-wide disk (see note). It will be irregular around the circumference — that's good, as all those irregularities will become crunchy goodness. The meat will shrink back down to bun-size as it cooks. Set the smashed meat sheet aside, replace the wax paper and repeat flattening process with the remainder of the meat balls, then refrigerate all of the smashed patties while you heat the oil.
Get a nice big Dutch oven and add enough peanut oilIf you prefer not to use peanut oil, safflower or canola are fine too. to fill it two inches deep. Install your candy/fry thermometer to the side of the pot and crank the heat to medium-high. Your thermal destination is 320°F.
Meanwhile, in a medium bowl, toss together the cheese, garlic powder, and paprika until all the powder has stuck to the cheese. (If the powder doesn't stick, let the cheese warm up a bit on the counter and toss again.)
Grate the cheese and toss with the paprika and garlic powder until all the powder has stuck to the cheese.
Heat the broiler to high and place the rack in the top third of oven. (This is a perfect time to use your toaster oven if you have one.) Arrange the bun halves cut side-up on a sheet pan. Spread a thin layer of mayonnaise on the bottom halves and mustard on the tops, then cover both with the cheese mixture and broil briefly to melt the cheese. I wouldn't walk away as this will happen pretty quickly, 30 seconds to 1 minute.
When the oil hits 320°F, remove the meat patties from the refrigerator. Cook one at a time by peeling the wax paper from one side of the patty then flipping it onto the spider and peeling off the other piece of paper. Gently lower the spider into the fat and jiggle until the patty floats free, taking care not to let the patty fold in half. Cook one minute — no more, no less. If you’re feeling ambitious, you can add a second patty to the oil, but watch the oil temperature and don't let it drop under 300°F.
Using the spider, remove the fried meat to a paper towel and drain briefly, then transfer right away to the bun bottom. Top with the pickles then the bun top. The ideal order: bread, mayonnaise, cheese, meat, pickles, cheese, mustard, then bread.
Consume or wrap in aluminum foil and hold for up to 30 minutes before serving.