In a 3-1/2- or 4-quart slow cooker combine eggs, cheese, spinach, artichoke hearts, milk, shallots, basil, prosciutto, garlic, and pepper. Cover and cook on low-heat setting for 4-1/2 to 5 hours.
Wrap tortillas in paper towels; heat in microwave oven on 100 percent power (high) for 1 to 1-1/2 minutes or just until warmed. Spread pesto over each tortilla to within 1 inch of edge. Evenly divide egg mixture among tortillas, breaking it up as necessary; roll up, tucking in ends. Serve burritos with warm marinara sauce.