Adjust rack to top position and preheat oven to 425 degrees. Wash and dry produce.

Cut potatoes into ½-inch-thick wedges. Halve, peel, and thinly slice onion.

Toss potatoes on a baking sheet with a large drizzle of oil, half the Fry Seasoning (you'll use the rest later), salt, and pepper. Roast on top rack until browned and tender, 20-25 minutes.

While potatoes roast, heat a large drizzle of oil in a medium, preferably nonstick, pan (large pan for 4 servings) over medium-high heat. Add onion and cook, stirring occasionally, until browned and softened, 8-10 minutes.

Add 1 tsp sugar (2 tsp for 4) and a splash of water. Cook, stirring occasionally, until onion is jammy and caramelized, 2-3 minutes more. Season with salt and pepper.

Turn off heat; transfer onion to a small bowl. Wash out pan.

While onion cooks, in a second small bowl, combine mustard, mayonnaise, and ketchup.

Form beef* into two thin patties, each about the shape of a bread slice. Season all over with remaining Fry Seasoning, salt, and pepper.

Heat a drizzle of oil in pan used for onion over medium-high heat. Add patties and cook until browned and cooked through, 3-5 minutes per side.

Turn off heat; transfer patties to a paper-towel-lined plate. Wipe out pan.

Evenly spread one side of each sourdough slice with special sauce. Top half the sourdough slices, sauce sides up, with patties, caramelized onion, and cheddar. Top with remaining sourdough slices, sauce sides down, to form sandwiches.

Melt 1 TBSP butter in pan used for patties over medium heat. Add sandwiches and cook until bread is golden brown and cheese melts, 3-4 minutes per side. (For 4 servings, you may need to work in batches, adding more butter as needed.)

Transfer patty melts to a cutting board.

Halve patty melts on a diagonal.

Divide patty melts between plates. Serve with potato wedges and any remaining special sauce for dipping.