Lightly brush the steaks with the oil on both sides and season with salt and pepper.
Preheat the grill.
Combine the cheese and the basil, set aside.
Sear the steaks for 3 to 4 minutes. Turn the steaks over and add a little of the cheese and basil mixture to each. Close the lid and continue grilling until the cheese is melted and the steaks are cooked to your desired doneness.
Remove from the grill and let rest for 3 to 5 minutes.
In a saucepan over medium heat, combine the butter and garlic and heat until the butter is melted and the garlic is sizzling, about 3 minutes. Remove from the heat. Spoon over the steaks and serve immediately.