Arrange frog legs in single layer in baking dish. Combine oil, lemon peel & juice, onion, parsley, salt, mustard and basil. Measure out 1/3 cup marinade, cover with plastic wrap and chill. Pour remaining marinade over frog legs, turning to coat. Cover with plastic wrap and chill, for 3 hours, turning occasionally.
Brush grill with vegetable oil. Drain frog legs and discard marinade. Grill frog legs at medium heat, covered, for 3 minutes. Turn frog legs, cover, and cook for an additional 3-4 minutes or until meat is no longer pink and begins to separate from the bones.
Combine reserved marinade with butter and garlic in saucepan. Cook over medium heat for 1 to 2 minutes or until hot and butter is melted, stirring frequently. Pour over cooked frog legs and serve.