Ingredients

• 2 tablespoons neutral vegetable oil, such as grapeseed
• 2 medium onions, thinly sliced
• 1/2 teaspoon kosher salt
• 1/4 teaspoon freshly ground black pepper
• 2 tablespoons tomato paste
• 3 tablespoons Worcestershire sauce
• 2 tablespoons white wine vinegar
• 2 tablespoons unsalted butter
• 1 tablespoon light brown sugar
• 1/4 cup white wine vinegar
• 2 tablespoons finely chopped shallot (about 1 small shallot)
• 1/2 cup (1 stick) unsalted butter, room temperature
• 1/4 teaspoon kosher salt
• 1 tablespoon finely chopped tarragon or parsley
• 1 1/2 pounds dry-aged steak (such as rib-eye or New York strip; 20% fat), coarsely ground
• Neutral vegetable oil, such as grapeseed (for grill)
• Kosher salt, freshly ground black pepper
• 4 slices white cheddar cheese
• 4 brioche or other high-quality hamburger buns
• Romaine or green leaf lettuce (for serving)