In a small bowl, combine the cheese, basil, mayonnaise, mustard, and salt and pepper to taste. Set aside.
With dampened hands, shape the beef into 4 patties, about 4 1/2-inches in diameter and 3/4-inch thick. Set aside.
In a very large skillet or griddle over medium heat, cook the bacon until crisp and golden, 8 to 10 minutes, turning halfway through. Transfer the bacon to a paper towel-lined plate and set aside.
Prepare a grill to medium-high heat and lightly oil the grate. Sprinkle both sides of the patties with the salt and pepper. Grill to desired doneness, about 4 minutes per side for medium. During the last minute, place the buns on the grill, cut side down, to lightly toast.
Arrange the bacon on the bun bottoms. Top with the lettuce, onion, patties, cheese mixture, and bun tops and serve.Copyright Jill Silverman Hough. All rights reserved. www.jillhough.com.