Let's be honest, everyone orders French onion soup for the gooey cheese topping and most of all the crunchy, crispy edges where the cheese browns and gets extra delicious. I intentionally drape a big piece of cheese over this burger so it melts on to the pan and caramelizes. You are welcome!
Melt 2 tablespoons of the butter in a large saute pan over medium heat. Add the bacon and cook until some of the fat has rendered and the bacon is just starting to crisp, about 2 minutes. Reduce the heat to medium low and add the onion, thyme, 1/4 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the bacon is very crisp and the onions are deep golden brown and caramelized, about 15 minutes. If the pan is getting too dark, add 2 tablespoons of water and scrape the brown bits from the bottom of the pan; continue to cook until all the water has evaporated.
Meanwhile, stir together the mayonnaise, whole-grain mustard and Dijon in a small bowl. Spread the remaining 2 tablespoons butter on the inside of each bun. Shape the ground beef into four 6-ounce patties, each about 1/2 inch thick. Use your thumb to make a deep depression in the center of each patty. Season both sides generously with salt and pepper.
Heat a griddle (preferably cast-iron) over medium-high heat and grease with some butter. Place the buttered buns on the griddle and cook until lightly toasted, about 1 minute. Move to a cutting board. Place the burgers on the griddle and cook, undisturbed, until the bottoms are well browned, 3 to 4 minutes. Flip the burgers and top with 1/2 cup of cheese each. It is ok if it spills over -- it will get crispy that way. Cover with a large heat-proof bowl and continue to cook until the cheese is completely melted and starting to crisp on the griddle, about 2 minutes. Uncover and continue to cook until the edges of the cheese are crispy, about 2 minutes. (This timing should result in medium-rare to medium burgers.) Transfer to the bottom buns.
Top the burgers with the caramelized onions and bacon. Spread the Dijonnaise on each top bun, then sandwich together and serve.