How to Make It
Preheat broiler with oven rack 6 to 8 inches from heat. Combine Champagne vinegar, water, 1 tablespoon of the sugar, and 1 teaspoon of the salt in a microwavable bowl; microwave on HIGH 1 minute and 30 seconds. Add mushrooms; chill, uncovered, 10 minutes.
Heat oil in a large cast-iron skillet over medium-high. Shape lamb into 4 (1/4-inch-thick) patties. Sprinkle evenly 1 teaspoon each of the salt and pepper. Cook patties until browned and done, about 4 minutes per side. Transfer to a plate lined with paper towels.
Add tomatoes and garlic to skillet; cook, stirring often, until garlic begins to brown, about 2 minutes. Add white vinegar and remaining 3 tablespoons sugar, 1 1/2 teaspoons salt, and 1/2 teaspoon pepper; cook, stirring occasionally, until tomatoes begin to burst, about 10 minutes. Using a large spoon, press tomatoes, and cook, pressing occasionally, until almost all of the liquid is evaporated, about 10 minutes. Transfer tomato mixture to a bowl; cool 5 minutes.
Place 1 patty on each bottom hamburger bun. Top each patty with 1/4 cup cheese. Broil in preheated oven 2 minutes. Top patties evenly with tomato mixture, mushrooms, and watercress. Cover with top buns.