In a small bowl, combine the mayonnaise, sun-dried tomatoes, lemon juice, garlic, sugar, and a grind of pepper. (The mayonnaise can be made up to 1 day ahead and refrigerated.)
Position a rack about 4 inches from the broiler and heat the broiler on high. Heat a broiler-safe cast-iron grill pan or skillet on the stovetop over medium-high heat until searingly hot (at least 5 and up to 15 minutes).
Put the Worcestershire sauce in a large bowl. Gently break up the meat and add to the bowl. Season with 1 tsp. salt and 1/2 tsp. pepper and combine gently with your hands, taking care not to overmix. Form into 4 equal 3/4-inch-thick patties. Using your thumb, make a depression in the center of each patty on both sides to keep them from bulging.
Put the patties in the hot pan and transfer to the broiler. Broil until the top looks cooked but not browned, and the bottom has grill marks if using a grill pan, or is browned if using a skillet, about 2 minutes; flip the patties (rearranging if your broiler heat isn’t even) and cook to your desired doneness, 1 to 2 minutes for medium rare (130°F).
While the burgers are cooking, split the buns if necessary and arrange them cut side up on a baking sheet. Transfer the cooked burgers to a plate. Broil the buns until lightly toasted, about 30 seconds.
Spread both halves of the buns with the mayonnaise. Put the burgers on the bottom halves and top with the arugula and bun tops. Serve.