Melt the butter in a large saute pan over medium-high heat. Add the onions and season liberally with salt and pepper. Cook until lightly caramelized and browned on the edges, 8 to 10 minutes. Turn off the heat, cover with a lid and reserve. Using a 1/4 cup dry measuring cup, divide the beef into 12 individual patties in the shape of hockey pucks. When scooping the meat, use your fingers to push the meat into the measuring cup, then lightly press the meat down so it is level. Lightly tap the measuring cup on a cutting board or other flat surface to release the meat from the cup. Place the patties on a large baking sheet. Smash each patty with the back of a large flat spatula, about 1/4-inch thick. Liberally sprinkle each patty on both sides with salt and pepper. Dredge each patty in the all-purpose flour.
Heat a two-burner griddle or electric griddle to medium heat. Add half of the canola oil, making sure to coat the entire griddle. Carefully place 6 of the dredged patties on the griddle. Cook until lightly browned, 1 to 2 minutes, pressing down lightly with the back of a spatula. Flip and add 1 slice of the American cheese on each burger patty. Cook until the meat is cooked to medium and the cheese is melted, pressing down lightly again with the back of a spatula, 1 to 2 minutes. Transfer the cooked burger patties to a large plate or baking sheet. Use the spatula to scrape any cheese that has melted on to the griddle and a dish towel to wipe it down. Add the remaining canola oil and repeat the cooking process with the last 6 meat patties, this time only adding American cheese to 2 of the patties.
To assemble, stack 1 plain beef patty on top of 2 cheese-topped patties on each toasted bottom bun. Evenly top with the sauteed onions and finish with a top bun on each burger. Serve each with a kosher dill pickle spear and French fries.