SPECIAL CHEESE BURGER.
8 ounces Comté cheese, shredded (about 2 cups), divided
4 brioche hamburger buns, split
½ pounds ground chuck
½ tablespoons canola oil, divided
1 large red onion, thinly sliced
2 teaspoons kosher salt, divided
¼ teaspoons black pepper, divided
¼ cup (2 ounces) dry red wine
¼ cup mayonnaise
Butter lettuce leaves, for serving
Preheat oven to 425°F. Line a large rimmed baking sheet with parchment paper. Arrange 4 (1/4-cup) mounds of cheese on prepared pan spaced 3 inches apart and 11/2 inches from edges. Flatten each mound into a 3-inch circle. Bake in preheated oven just until edges of cheese rounds look dry and lacy, 6 to 8 minutes. Remove from oven. Working quickly, place 2 split buns, cut sides down, on top of cheese rounds, and press down. Let stand, undisturbed, 5 minutes. Slide parchment with buns off baking sheet, and let buns stand on parchment, undisturbed, until ready to serve. Line baking sheet with fresh parchment; repeat with remaining cheese and buns.
While cheese rounds bake, shape beef into 4 (4-inch-wide) patties. Make a slight indentation in center of each patty using your thumb; set aside.
Heat 1 1/2 tablespoons oil in a large cast-iron skillet over medium-high. Add onion, 1/2 teaspoon salt, and 1/4 teaspoon pepper; cook, stirring often, until slightly softened and browned around edges, about 8 minutes. Reduce heat to medium-low; cook, stirring often, until softened and lightly browned, 6 to 8 minutes. Add wine; cook, stirring constantly, until wine is absorbed, about 1 minute. Transfer onion mixture to a small bowl, and wipe skillet clean.
Sprinkle patties with remaining 1 1/2 teaspoons salt and remaining 1 teaspoon pepper. Heat remaining 1 tablespoon oil in skillet over medium-high. Working in batches, cook patties to desired degree of doneness, about 3 minutes per side for medium. Transfer patties to a plate.
Spread 1 tablespoon mayonnaise over cheese on each bottom bun half. Top each with 1 burger patty, 1/4 cup cooked onions, and lettuce. Cover with top buns, cheese sides down.