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.
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.
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.
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.
The wagyu patties can be shaped ahead of time and divided by wax paper, then covered with cling wrap in the refrigerator until cooking time.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.
Recipe courtesy of Twisted Root Burger Co., Dallas, TX