Halve the artichokes and slice each half into 4 segments. Wipe dry with paper towels.
Blanch the fava beans for 5 minutes in boiling water, then plunge into cold water. Drain and reserve.
In a heavy skillet, heat the oil over medium heat and saute the artichokes, onion, and fava beans. When the vegetables are tender, about 15 minutes, add the minced garlic and saute briefly.
With a slotted spoon, remove the vegetables from the skillet, leaving as much of the oil in the skillet as possible. Pass the oil through a fine sieve, wipe the skillet with paper towels, and pour the oil back into the skillet.
Beat the eggs lightly. Heat the oil over low heat and pour the eggs into the skillet. Add the sauteed vegetables and sausage slices, and season with salt and pepper. Stir the egg mixture with a fork for a few seconds, until the omelet begins to thicken. Continue cooking for about 1 minute, or until the omelet is almost firm and starts to brown on the bottom. Carefully slide the omelet onto a plate, invert the skillet over it, and turn over. Saute the omelet for a few seconds on the second side.
Transfer to a serving dish, sprinkle with chopped parsley, and serve immediately.