Remove corn husks but leave the core attached.
Drizzle the garlic bulb with oil and wrap in aluminum foil. Grill garlic for about 15 minutes.
Brush 3 tablespoons of butter all over corn cobs and grill, turning so each side of the corn gets a slight char, about 15-20 minutes.
Remove corn and garlic from the grill, and cover corn to keep it warm.
Open the garlic foil and let cool, then add 4 tablespoons of butter, garlic cloves, salt, and pepper to a small bowl and mash and mix well.
Spread garlic butter over corn and sprinkle parmesan and basil over the top.