In a large saute pan, heat the butter and/or olive oil, garlic, and basil. Stir together with a wooden spoon.
Once melted and combined, spread the basil onto one side of each slice of bread, then place each slice of bread, basil side down, into the saute pan.
Once browned, add the cheese slices to one of the slices of bread, then top with the remaining slice of bread. Cook until the cheese melts, flipping sandwich as needed.
When cheese is melted, remove from heat and add avocado slices to the sandwich. Serve and enjoy immediately!