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