Burger with Caramelized Onions, Asiago, and Basil Mayonnaise

Ingredients

1 tablespoon olive oil
3 cups thinly sliced red onion
2 teaspoons sugar
3/4 teaspoon salt, divided
1/8 plus 1/4 teaspoon ground black pepper, divided
1 tablespoon balsamic vinegar
7 tablespoons mayonnaise
1 tablespoon Dijon mustard
2 tablespoons finely chopped fresh basil
1 1/2 pounds 80% lean ground beef
4 (1-ounce) slices Asiago cheese
4 brioche or similar hamburger buns, split horizontally
1 cup baby romaine lettuce
8 tomato slices

Directions

1. Heat the oil in a 10-inch nonstick skillet over medium heat. Add the onions, sugar, 1/4 teaspoon of the salt, and 1/8 teaspoon of the pepper.
2. Reduce the heat to medium and cook, stirring occasionally, until the onions are golden brown and about 16-18 minutes. Stir in the vinegar and cook 1 minute longer; remove from the heat.
3. Meanwhile, combine the mayonnaise, mustard, and basil in a small bowl; mix well and reserve.
4. Form the ground beef into 4 (4-inch diameter) patties, sprinkle with the remaining 1/2 teaspoon of salt and 1/4 teaspoon of pepper and make a 1/4-inch depression with your thumb in each. (This helps the burger cook more evenly). Heat a nonstick grill pan over medium, add the burgers and cook 4 minutes per side for medium-rare. Top each burger with 1 slice of cheese and loosely cover with aluminum foil. Cook 1 minute longer until cheese melts.
5. To assemble, place top 1/4 cup baby romaine lettuce, 2 tomato slices, burger, and 1/4 of the onions on each bun bottom. Spread the cut side of the top half of each bun with 1/4 of the mayonnaise mixture and set on top of each burger. Serve immediately.