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 the meat mixture into 4 equal portions, gently shaping each into a 1/2-inch-thick patty. Press a nickel-sized indentation in the center of each patty. Sprinkle evenly with the remaining 1/4 teaspoon salt.
Heat a large cast-iron skillet over medium-high heat. Add the remaining 1 teaspoon oil to pan; swirl to coat. Add patties, and cook for 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 shallots. Spread about 2 teaspoons mayonnaise on top half of each bun; place on top of burgers.
In all fairness I could not give this five stars because I substituted Jarlsberg cheese for the cheddar so therefore am not reviewing as written. That said, every other part of it was outstanding and the shallots are such a different taste than onions! My husband is particular about his burgers and said it was "the best burger of the summer". :) We did add a sliced tomato as one other reviewer recommended. We will make this again with friends over.
Just a damn fine burger. Adding a slice of fresh summer tomato makes it even better. I grilled it at med-hi (3 second hand count) for 3 minutes per side, covered, and it came out perfect. Given the overall ease of prep, this will be a go to burger for grilling season.
If you love a good burger this is for you!! I prepared everything exactly as written in the recipe. When I'm watching my calories, what I love is flavorful foods that totally satisfy. This burger completely satisfied that burger craving. Eat it slowly and savor all the interesting favors. Had a side of fresh mixed berries with the burger. Delizioso!