Drizzle the garlic bulb with oil and wrap in aluminum foil. Turn one side of the grill down to medium heat and add the garlic to the grill 15 minutes before the corn.
Add the cobs to the grill and grill until each side gets a little char, turning often and checking so they don’t burn. About 10 to 15 minutes.
Remove both the corn and the garlic. Open the foil to allow the garlic to cool so you can handle it.
Cover the corn with foil to keep warm while you make the butter.
To a bowl add the softened butter, the cooled garlic cloves, salt and pepper. Mash and mix well.
To a plate add the Parmesan cheese and basil, mix well.
Spread the butter all over the corn cobs, then roll in the Parmesan and basil. Serve warm.