In an ovenproof frying pan, heat the butter or oil and sauté onion; when onion is nearly translucent, add garlic. Add vegetables and sauté until tender but not browned. Add potato. Stir to combine; warm until heated through.
Beat eggs and add herbs, cayenne, salt and pepper. Pour over vegetables. Spread basil and cheese over the top. Cook at medium low heat on stovetop for about 15 minutes or until the eggs are just set; place the pan under the broiler for a few minutes, until the eggs are fully set and cheese is browned.
Cut into wedges and serve.