Light a grill. In a bowl, mix the ground meats and gently form into four 4-inch patties, about 1 1/2 inches thick. Make a 1/2-inch-deep indentation in the center of each patty, then rub lightly with the oil and season generously with salt and pepper.
In a processor, blend 4 ounces shredded sharp cheddar, 1/2 teaspoon dry mustard, 1/4 teaspoon pepper, 1 tablespoon mayonnaise and 2 tablespoons chopped pimentos. Stuff the burgers with the pimento cheese before grilling.
Grill the burgers over high heat, turning once, about 6 minutes for medium-rare; transfer to the buns, top with the relish and serve.