Soak fries (optional but best). Place cut potatoes in a bowl of cold water for 20 to 30 minutes. Drain, rinse, and dry very well with a towel.
Roast fries. Heat oven to 450F. Toss dried fries with olive oil, salt, pepper, and optional seasonings. Spread on a large sheet pan in a single layer (use two pans if crowded). Bake 18 minutes, flip, then bake 10 to 15 minutes more until deeply golden and crisp. Salt again to taste.
Make burger sauce (optional). Stir sauce ingredients together and set aside.
Shape patties. Divide beef into 4 equal portions and gently form into patties about 3/4-inch thick. Press a small dimple in the center of each patty. Season both sides generously with salt and pepper.
Cook burgers. Heat a skillet or grill over medium-high heat. Cook patties 3 to 5 minutes per side (depending on thickness) until cooked to your liking. Add cheese in the last minute and cover briefly to melt.
Toast buns. Butter cut sides and toast in the skillet 1 to 2 minutes until golden.
Assemble. Add sauce and toppings, then serve immediately with hot fries.