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Heat a nonstick skillet over medium-low heat. Add 2 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 1 minute. Remove from heat; keep warm.
Gently combine garlic and beef. Divide meat mixture into 4 equal portions, gently shaping each into a 1 ⁄2 -inch-thick patty. Press thumb in center 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 1 teaspoon oil to pan; swirl to coat. Add patties, and cook 3 minutes on each side or until desired degree of doneness. Top each patty with 2 tablespoons cheese; cover and cook 1 minute or until cheese melts.
Place 1 ⁄4 cup arugula on bottom half of each bun; top with 1 patty and one-fourth of shallot mixture. Spread about 2 teaspoons mayonnaise on cut side of top half of each bun; place on top of burgers.