This is a Basque specialty, and quick to make if the pipérade mixture has been prepared in advance. As the omelette is not folded, and is served in its cooking vessel, it is not a disaster if it sticks a little on the bottom. You may therefore cook the eggs in a low, glazed pottery dish, or a fancy skillet.
Heat the oil and butter until hot, then brown the ham slices lightly on both sides. Set them aside, and reheat just before using them at the end of the recipe. In the same pan, cook the onions and peppers slowly, covering the skillet, until they are tender but not browned. Season to taste with salt and pepper. Stir in the garlic and cayenne. Lay the tomatoes over the onions and sprinkle with salt. Cover and cook slowly for 5 minutes. Uncover, raise heat, and let cook for a few minutes, shaking the pan occasionally until the juice from the tomatoes has almost entirely evaporated. Season to taste, and reheat just before using.
Heat the remaining oil or butter in a skillet or in a shallow, fireproof serving dish. Meanwhile, beat the eggs lightly, and season with the salt and pepper. When the oil is very hot, pour in the eggs. Stir rapidly with a fork, until the eggs have just set into a creamy mass. Remove from heat and spread over them the hot pipérade over them, mixing a bit of it delicately into the eggs. Lay the warm ham strips over the pipérade. Sprinkle with the herbs and serve immediately.