1 tablespoon olive oil, divided
1 pound 90 percent lean ground beef
2 cups thinly sliced shallots
1/2 cup shredded sharp cheddar cheese
1/2 teaspoon kosher salt, divided
1 cup arugula
1 tablespoon white wine vinegar
4 hamburger buns toasted
2 cloves garlic, minced
3 tablespoons light mayonnaise
Heat a nonstick skillet over medium-low heat. Add two teaspoons oil; swirl to coat. Add shallots and 1 ⁄4 teaspoon salt; cook 15 minutes or until golden brown, stirring occasionally. Stir in vinegar; cook one minute. Remove from heat; keep warm.
Gently combine garlic and beef. Divide meat mixture into four equal portions, gently shaping each into a 1 ⁄2 -inch-thick patty. Press thumb in centre of each patty, leaving a nickel-sized indentation. Sprinkle evenly with 1 ⁄4 teaspoon salt.
Heat a large cast-iron skillet over medium-high heat. Add one teaspoon oil to pan; swirl to coat. Add patties, and cook three minutes on each side or until desired degree of doneness.
Top each patty with two tablespoons cheese; cover and cook one minute or until cheese melts.
Place 1 ⁄4 cup arugula on bottom half of each bun; top with one patty and one-fourth of shallot mixture. Spread about two teaspoons mayonnaise on cut side of top half of each bun; place on top of burgers.