1. In a food processor, coarsely chop mint leaves, pine nuts, garlic, and 1/4 teaspoon each of salt and pepper. With machine on, pour 1/3 cup of olive oil through feed tube and puree until smooth. Add cheese and process about 5 seconds to mix well. Use pesto immediately or refrigerate up to 24 hours.
2. Prepare a medium-hot fire in a barbecue grill. Pound chicken between pieces of waxed paper to an even 1/4-inch thickness. Season with remaining salt and pepper. Spread pesto over chicken, then roll up to enclose pesto. Press gently to flatten rolls slightly and help chicken adhere to itself. Rub rolls with remaining 1 tablespoon oil.
3. Grill chicken, turning once or twice with a spatula, until white through-out, 13 to 15 minutes.