Mix mayonnaise, pickle relish, ketchup, minced onion, olive oil, sugar, vinegar, lemon juice, dry mustard, paprika, salt, and pepper in a bowl to make sauce. Refrigerate until flavors combine, at least 30 minutes.
Preheat a large grill pan over medium heat. Cook hamburger patties, turning occasionally, until browned, 10 to 15 minutes. An instant-read thermometer inserted into the center should read at least 160 degrees F (71 degrees C). Chop patties into bite-sized pieces.
Toss chopped hamburger patties, salad greens, Cheddar cheese, diced onion, and pickle chips in a large bowl. Top with sauce before serving.