Step 1

Combine onion, 1 tablespoon oil and 1 tablespoon ketchup in a small saucepan. Cover and cook over medium heat, stirring often, until the onion is softened, 5 to 7 minutes. Transfer to a medium bowl.

Step 2

Combine the remaining 1 tablespoon ketchup, mayonnaise, relish and vinegar in a small bowl. Set aside.

Step 3

Add beef, Worcestershire and pepper to the onion mixture and gently combine without overmixing. Form into four 4-inch patties.

Step 4

Heat the remaining 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add the burgers, reduce heat to medium and cook, turning once, until just cooked through, 5 to 7 minutes per side. Assemble the burgers on buns with the ketchup mixture, tomato and lettuce.