French Onion Burgers

Let 4 tablespoons unsalted butter sit at room temperature until softened. Meanwhile, heat the grill, make the patties, and cook the onions.

Heat an outdoor gas grill for medium, indirect heat (about 350°F) or until you can hold your hand an inch over the grates for 3 to 4 seconds. For a gas grill, light 2 burners to medium and leave one burner off. For a charcoal grill, arrange 1/2 chimney of unlit lump charcoal on one half of the lower grate. Then light 1 chimney full of lump charcoal. When the coals are white and ashy, pour onto the unlit charcoal, arranging the coals to create a high heat and a low heat cooking zone (direct and indirect grilling).

Halve and thinly slice 1 1/2 pounds yellow onions (about 4 cups). Pick the leaves from 8 fresh thyme sprigs until you have 1 tablespoon.

Place 1 1/2 teaspoons of the kosher salt, 2 teaspoons onion powder, 1/2 teaspoon garlic powder, and 1/4 teaspoon black pepper in a large bowl and stir to combine. Add 1 1/4 pounds ground beef and mix with your fingertips until the seasoning is evenly distributed. Do not overmix. Divide into 4 (5-ounce) portions. Use your hands to gently flatten each portion into a patty about 3/4-inch thick and press your thumb into the center to create a shallow divot. Let the burgers sit at room temperature until the onions are finished cooking, about 30 minutes.

When the grill is ready, place 2 tablespoons of the butter in a 12-inch cast iron or other grill-safe, heavy bottomed skillet. Place over medium, direct heat until melted. Add the onions, thyme, remaining 3/4 teaspoon kosher salt, and remaining 1/2 teaspoon black pepper. Cook, stirring frequently with a wooden spoon, until the onions have reduced by half, are very soft, and deep golden-brown, 20 to 25 minutes on a charcoal grill and 50 to 55 minutes on a gas grill. Cover between stirs if using a gas grill, or cook uncovered if using a charcoal grill. Move the pan temporarily to indirect heat (over an unlit burner or away from the pile of coals) if the onions start to burn.

Position the skillet over direct heat. Pour 1/4 cup dry white wine over the caramelized onions and scrape up any browned bits on the bottom of the pan. Continue to stir until the wine is evaporated, about 20 seconds. Remove the pan from the heat, add 1 teaspoon balsamic or sherry vinegar, and stir to combine.

Place the patties on the grill in a single layer over direct heat. Cook uncovered until grill marks form on the bottom and the burgers release easily from the grill grates, 2 to 4 minutes. Flip and cook until grill marks appear on the second side, about 2 minutes more. Move the patties to indirect heat and continue to grill 3 to 5 minutes more for medium-rare burgers (120 to 125°F), 6 to 8 minutes more for medium (130 to 135°F), or 8 to 10 minutes more for medium-well burgers (140 to 145°F).

In the last 2 to 3 minutes of cooking, place 2 ounces Gruyère cheese on top of the patties. Spread the remaining 2 tablespoons butter over the cut sides of 4 hamburger buns. Arrange buns, buttered-side down over direct heat and grill until lightly toasted, turning once, about 10 seconds.

Place a small handful of baby arugula, if using, on each bottom bun. Top each bottom bun with 1 burger patty. Divide the caramelized onions evenly among the burgers and grind some more black pepper on top. Close with the top buns.