Basic burgers are boring and bland. I didn't used to think that, but sure do now. This burger may be the best we've ever had. Three steps here:
Saute onion rounds in olive oil; sprinkle with 3/4 teaspoon coarse salt and pepper. toss in 1/2 cup vinegar and cool until onions are tender, about 20 minutes. Can be made 3 days ahead. Cover; chill.
For chipotle ketchup:
Mix ketchup and canned chipotle chiles in small bowl. Season with salt. Can be made 3 days ahead. Cover and chill.
Shape lamb into 1/2 pound, 1/2-inch thick patties. Sprinkle patties on both sides with coarse salt and pepper. Prepare stove top skillet (preferably cast iron), and Pplace burgers on skillet. Cook burgers until bottoms start to darken and juices rise to surface, about 3 minutes per side. Top with onions and cheese, and place burgers under the broiler until the cheese is melted and bubble. While the burgers are in the broiler, also Broil or toast the sourdough slices, then spread em with ketchup. Top with burgers, lettuce, and anything you may like such as tomatoes, pickles, etc. Serve with remaining ketchup to pass.