Heat a large saute pan, then toss in the onion and garlic slices with 2 tablespoons oil and a generous pinch of salt. Cook over medium heat for 2 to 3 minutes, then add the Marsala, brown sugar and mustard and cook, stirring occasionally, until the onions are caramelized and tender, 20 to 25 minutes. (Do not rush this step, because properly caramelized onions make this burger top-notch!) Stir in the vinegar and cook to blend in, 1 to 2 minutes. Set aside.
For the burgers:
In a medium bowl, break up the meat and spread it up on the sides of the bowl so that the seasoning really permeates. Sprinkle the meat with salt and pepper. Gently mix in the Marsala, soy sauce and Worcestershire. Do not overmix. (The more you work the meat, the tougher the burger.) Form the meat into 4 even patties that are about 3/4-inch thick and transfer to a baking sheet.
Use a spoon to gently scoop out a little meat and make a hole or pocket in the center of each patty that does not go all the way through to the bottom. Place about 1/4 cup Gruyere in the hole of each patty. Flatten the piece of meat you scooped out and press it over the cheese so that it's hidden in the middle of the patty. Turn the patty on the other side that it is weighing down on the seam. Repeat with the remaining patties. Refrigerate for at least 30 minutes and up to 6 hours.
Preheat the oven to 400 degrees F.
Heat a large skillet over medium-high heat and add the remaining 4 tablespoons oil. Sprinkle the patties lightly with salt. When the oil begins to smoke lightly, arrange the patties in a single layer, cheese-pocket-side down, at a distance from one another. Cook the patties, undisturbed, until the bottom browns, 3 to 5 minutes. Turn the patties on their other sides and let the bottoms brown again, 3 to 5 minutes. Lower the heat and add the stock to the skillet and cover. Cook on low heat until the patties are cooked through, 8 to 10 additional minutes. Remove the cover, shut off the heat under the skillet and let the patties rest with the heat off.
Arrange the bun halves in a single layer on a baking sheet and sprinkle each bun half with about 1/4 cup Gruyere. Place the baking sheet in the center of the oven and toast until the buns are golden brown and the cheese is bubbling and melted, 5 to 8 minutes. Season with salt and pepper.
Arrange the bun bottoms on a flat surface. Mix the onion jam and remaining cup Gruyere into the liquid in the skillet. Taste for seasoning. Spoon some of the onions and liquid on each bun bottom and top with the patties and more onions. Top with a few slices of pickle and season the bun tops with a pinch of salt. Close the burgers and serve.
This burger defies classic burger rules: The bun is deliberately soggy and the cheese is Gruyere instead of the mainstay American. It's really like a French onion soup hamburger. What kinds of meat? I choose an 85/15 blend if I buy pre-ground at the supermarket. If I have a butcher grind for me, I like sirloin and short rib, about equal parts. You could also get away with this burger on some deeply toasted sourdough bread in place of the buns…