Preheat oven to 425°F.
Place 2 tablespoons vinegar and about 3 cups water in a bowl. Cut the peeled potatoes first into planks then into thin fries. Add fries to vinegar bath and soak 20 minutes.
Drain fries and dry, then spread them out on a large baking sheet in a single layer. Sprinkle fries with remaining 2 teaspoons vinegar and spray with cooking spray. Season with salt and roast 20 minutes; turn fries over and roast 20 more minutes. Remove from oven and salt to taste.
While the fries are cooking, in a skillet over medium heat, melt butter for the onions. Add onions and bay, and season with salt and both peppers. Cook onions 15-20 minutes until very soft and light caramel in color. Add 1 cup of water and cook 10 minutes longer until onions are a deep caramel color and the water has been absorbed.
In a bowl, combine sour cream, ketchup, relish and Worcestershire sauce.
Heat a cast-iron skillet over medium-high to high heat with oil, 1 turn of the pan. Divide the meat into four portions and loosely roll each portion into a ball. Add balls to the hot pan and smash them into patties with a metal burger spatula. Season with salt and pepper, and cook 3 minutes then flip and top each patty with 2 slices of cheese. Cook burgers 2-3 minutes more, tenting with foil to melt the cheese.
To build the burgers, slather sauce on the bun bottoms and top with lettuce, pickles, a burger patty and some sweet onions. Set bun tops in place and serve.