Step 1

Heat oil in a skillet over medium-high heat. Add the onion and saute until golden brown and caramelized, 5 to 6 minutes. Add the thyme and mix well. Set aside on low heat until ready to serve.

Step 2

Divide wagyu into 4 equal patties (see Cook's Note). Spread 1 tablespoon Dijon mustard on one side of each patty. Place the patties, mustard-side down, in a large skillet over medium-high heat. Sprinkle with salt and pepper to taste. Cook until the mustard sides of the patties have cooked through and become a caramel brown color, then flip the patties over and cook until the centers of the patties have reached the ideal temperature for diners' tastes, using an instant-read thermometer. (Without a thermometer, a patty that is firm to the touch in the center but not tough and has a thin line of pink through the middle is around medium-well.) Remember, you still have to melt the cheese on top, so you may want to cook a little under your normal preference at this stage, because it will continue to cook while the cheese melts.

Step 3

Add 1 ounce (1/8 cup) sauteed onions to the top of each wagyu patty. Add 1 slice of Gruyere on top of each serving of onions. Cover the skillet for about 30 seconds to allow the steam to melt the cheese.

Step 4

Butter brioche buns and lightly toast using a separate skillet, if desired. Place a patty on each brioche bun. Serve with lettuce and tomato on the side for garnish, along with desired condiments.