Using a bread knife, carefully slice off the tips of the artichokes, remove all of the tough outer leaves, and using a paring knife peel the base. Cut each artichoke into quarters and remove the choke, as well as any prickly tips that may have formed on the inner leaves. Place in water with lemon juice. Drain. Cut each of these quarters into halves, to make wedges.
In a medium saucepan, heat 4 tablespoons extra-virgin olive oil, add onions, carrots, olives, capers, chiles, and drained artichokes. Season with salt and pepper. Toss to caramelize. Add more olive oil, season, add tomato paste and let ingredients bubble rapidly. Add anchovies. During the last 5 minutes add tomatoes. Add red wine vinegar and sugar and reduce 1 minute.
Serve hot or at room temperature with a salad