Preheat grill to medium-high (or see Stovetop Variation).Advertisement
Combine onion, oil and 1 tablespoon ketchup in a medium saucepan. Cover and cook over medium-high heat, stirring often, until the onion is softened, 5 to 7 minutes. Reduce heat to medium-low, uncover and cook, stirring occasionally, until very soft, 5 to 8 minutes more. Transfer to a medium bowl and let cool for a few minutes.
Meanwhile, combine the remaining 1 tablespoon ketchup, mayonnaise, relish and vinegar in a small bowl. Set aside.
Add beef, Worcestershire (or steak sauce) and pepper to the onion and gently combine without overmixing. Form into 4 patties, about 3/4 inch thick.
Oil the grill rack (see Tip). Grill the burgers, turning once, until an instant-read thermometer inserted in the center registers 165 degrees F, 4 to 5 minutes per side.
Assemble the burgers on toasted buns with the ketchup-mayonnaise sauce, tomato slices and lettuce.
Make Ahead Tip: Prepare ketchup-mayonnaise sauce (Step 3) and refrigerate for up to 1 day.
Stovetop Variation: Coat a nonstick pan, preferably cast-iron (or a grill pan), with cooking spray and heat over medium-high heat for 1 to 2 minutes. Add burgers, reduce heat to medium and cook, turning once, until an instant-read thermometer registers 155°F (for pork and bison) or 165° (for beef or chicken), 4 to 5 minutes per side.
Tip: To oil a grill rack, oil a folded paper towel, hold it with tongs and rub it over the rack. (Do not use cooking spray on a hot grill.)
1 1/2 starch, 1 vegetable, 3 lean meat, 1 fat