Slice bread on the diagonal so you have longer slices, into 8 slices for 4 sandwiches. Place olive oil and garlic in small saucepan and heat only till it's just warm and garlic infuses into oil. Remove from heat. Butter inside of bread slices. Place ham on microwavable dish and heat through. Blot any juices that lay on the plate so it doesn't make sandwiches soggy.
Preheat large skillet or griddle on stovetop on medium heat. Put ham on buttered bread, then basil leaves, then cheese. Brush garlic-infused olive oil on outside of sandwiches and place in preheated skillet or griddle. Lightly brown on each side.