Finely chop the large onion(s). Add to a large skillet with the butter over medium high heat. Saute for 6-8 minutes, until fragrant and beginning to brown. Add ¼ cup water and stir until water has evaporated. Repeat this until the onions are deeply caramelized, about 10-15 minutes.
Combine all the remaining ingredients together in a large bowl. Mix until well combined. Stir in ½ of the sautéed onions. Season with salt and black pepper to taste.
Peel the potatoes and slice into ¼" thick fries. Place the sliced fries in a large bowl with hot water and vinegar for 20 minutes, or boil for 10 minutes.
Drain the fries and pat dry. Let steam dry 5 minutes.
Bring the cooking oil in a large saucepan over medium high until glistening, about 8-10 minutes.
Working in small batches, add the fries to cook for 3-4 minutes, until golden. Remove with a slotted spoon to a paper towel lined plate. Repeat with remaining fries.
Still working in small batches, add the fries back into the hot oil to fry another 30 seconds and up to 2 minutes, until deeply golden and crispy. Remove with a slotted spoon as before.
Garnish with salt while still a little oily and hot.
Divide the ground beef into 4 equally-sized pieces. Roll into balls. Working one burger patty at a time, add to a greased pan over medium high heat. Place a square of parchment paper on top and smash down with a pan lid or burger smasher.
Cook for 2 minutes, then flip, add a slice of cheddar cheese, and cook another 1-2 minutes. Repeat with remaining ground beef.
On each toasted burger bun, add a generous slather of the secret sauce. Layer the bottom bun with a tomato slice seasoned with salt and black pepper, lettuce, the smashed cheeseburger, and sautéed onions. Add the top burger bun.
For the fries, add them to a baking pan and top with the sliced cheddar cheese. Broil on HIGH heat in the oven for 1-2 minutes, until cheese is melted. Top with the secret sauce and more onions. Enjoy immediately!
