Position an oven rack in the upper third of the oven and preheat to 475 degrees F.
Heat 2 teaspoons of the oil in a medium cast-iron skillet over medium-high heat. Add the onions and cook until softened, 3 to 5 minutes. Add the ground beef, season with 1/2 teaspoon salt and a pinch of black pepper and cook, breaking up the meat, until browned, 5 to 7 minutes. Drain off the fat and transfer the meat mixture to a medium bowl. Stir in the ketchup, mayonnaise and mustard.
Wipe the skillet clean, add the remaining 2 teaspoons oil and heat over medium-high heat. Sprinkle the pork chop liberally with salt and pepper. When the oil starts to shimmer, add the pork and cook until browned on both sides and just cooked through, 3 to 4 minutes per side. Rest for 5 minutes, then slice very thinly against the grain.
Line a sheet pan with parchment and rub with 1 tablespoon of the butter. Arrange 8 of the tortillas on top of the parchment, overlapping as needed. There should be a 3- to 4-inch overhang of tortillas on all sides. Position the sheet pan so one of the long edges is facing you.
To make the patty melt: In the top left quadrant of the tortillas, arrange half the American cheese, then top with the meat mixture and remaining American cheese.
To make the happy waitress (bacon tomato grilled cheese): Reserve 1/2 cup of the Cheddar and 1 strip of bacon for topping. In the top right quadrant, sprinkle half of the remaining Cheddar on the tortilla base, then top with the remaining 4 strips of bacon, the sliced tomato and remaining Cheddar.
To make the Cuban: Reserve 1/2 cup of the Swiss cheese from one of the packages for topping. In the bottom left quadrant, sprinkle half of the remaining Swiss from the first package on the tortilla base, then top with the ham, sliced pork, pickle slices, a drizzle of mustard and the remaining Swiss cheese from the first package.
To make the Reuben: In the bottom right quadrant, sprinkle half of the second package of Swiss cheese on the tortilla base, then top with the corned beef, sauerkraut, a drizzle of Russian dressing and the remaining Swiss cheese.
Put the last tortilla in the center and fold the edges of the bottom tortillas up and over to create a closed package. Brush the top of the tortillas with 1 tablespoon of the butter. Place a piece of parchment over top, then place a sheet pan and heavy skillet on top of the parchment to weigh the sandwich down.
Bake for 10 minutes. Remove the heavy skillet, top sheet pan and piece of parchment. Brush the top with the remaining tablespoon of butter and sprinkle each quadrant with the appropriate topping: for the patty melt (top left), sprinkle with the sesame seeds; for the happy waitress (top right), sprinkle with the 1/2 cup reserved Cheddar and crumbled remaining bacon strip; for the Cuban (bottom left), sprinkle with the 1/2 cup reserved Swiss cheese; and for the Reuben (bottom right), sprinkle with the caraway seeds. Return to the oven and bake until browned, 5 to 7 minutes more. Slide off the pan onto a cutting board. Cool 5 minutes before slicing and serving.
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