Mix the ground beef, salt and pepper in a large bowl and form the mixture into 8 equal-size balls. Press a cube of desired cheese into the center of each ball and cover completely with meat. Form the balls into hamburger patties, about 1/4-inch thick. Grill burgers until desired doneness. Top with lettuce, tomatoes, onions and pickles as desired on a toasted bun.
Place the drained pimientos in a blender or food processor and puree until smooth. Using an electric mixer, combine the pimientos and cheese, beating until smooth. Beat in the mayonnaise.