Preheat oven to 400°F Cook bacon in the oven on a baking sheet with sides, turning strips over once, 12 minutes, or until crisp. Drain on paper towel.
Place half the cheddar cheese on half the bread slices. Neatly arrange tomato slices, bacon strips, and avocado slices on top of cheese, then top with remaining cheese slices and bread.
Heat 2 large heavy skillets or a griddle over medium-low heat. Carefully spread both sides of each sandwich with butter. Grill sandwiches, pressing down frequently with the back of a spatula, until bread is crisp and toasted to an even golden color and cheese is melted, about 2 minutes per side. Serve immediately, cutting sandwiches diagonally into halves.