.For the CARAMELIZED ONIONS:
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
For the HERB BUTTER
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
And for the BURGER:
1 1/2 pounds dry-aged steak (such as rib-eye or New York strip; 20% fat), coarsely ground
Plain vegetable oil, such as grapeseed (for grill)
Kosher salt, freshly ground black pepper
4 slices white cheddar cheese
4 high-quality hamburger buns
Romaine or green leaf lettuce (for serving)