Place corn on grill and cook about 1 minute per side, or until ears reach a dark brown color. Rotate as necessary to achieve even charring.
Remove corn and immediately brush with butter, coating each ear evenly. Season with salt and pepper, place on serving plate, and drizzle basil aioli on cooked corn. Sprinkle cheese on top, garnish with basil leaves, and serve.
Preheat oven to 300 degrees.
Cut garlic heads in half lengthwise and coat with oil. Wrap garlic in aluminum foil and place packet on baking tray. Place in oven and cook until garlic is soft, about 1 hour. Let cool.
Once cool, remove soft cloves from brittle skin and mash cloves with form. Set aside 3 tablespoons (reserve any extra for another use).
Place garlic with remaining ingredients in blender. Blend until smooth.