Sliced pickled jalapenos, crushed tortilla chips and pico de gallo, for garnish (if using Homemade Queso)
Heat your griddle or large cast-iron skillet over high heat. Add the oil and let heat until it begins to shimmer.
Shape the ground beef with your hands into 4 round patties about 3/4 inch thick; season each burger on both sides with salt and pepper. Make a divot in the center of each burger. Add the burgers to the griddle or skillet, divot-sides up and cook, flipping once, until brown on both sides and cooked to medium doneness, about 8 minutes, total.
IF USING SLICED CHEESE: Top each burger with 2 slices of the cheese. Carefully, add 1/4 cup of water to the skillet and immediately cover tightly with a lid. If using a griddle, squirt a bit of water from a squeeze bottle then cover 1 burger at a time with a melting dome or metal bowl. Count to 10. Lift the lid or dome. If the cheese is totally melted, place the burgers on the bun bottoms; if not, cover again and count 5 more seconds. Top with lettuce, tomato, onion and pickles, potato chips, Dijon mustard and ketchup, if desired.
IF USING QUESO SAUCE: Place each burger on a bun bottom and top with a few tablespoons of the queso sauce. Top with the jalapenos, tortilla chips and pico de gallo.
Cover the burgers with the bun tops and serve immediately.
Melt the butter in a medium saucepan over high heat Add the flour and cook, whisking, until pale blond, about 1 minute. Slowly whisk in the 2 cups of hot milk, bring to a boil, then reduce the heat to medium and cook, whisking constantly, until thickened and the raw flour taste has cooked out, about 5 minutes.
Remove the saucepan from the heat and whisk in all 3 cheeses until smooth. Whisk in more hot milk if the queso is too thick. Season with salt, black pepper and cayenne.