In a large skillet, heat oil and saute onion with garlic. Simmer until onions become translucent. Add other vegetables and herbs. Cook 15 to 20 minutes, or until all ingredients are soft.
Mix eggs and milk. In a large skillet, heat butter, then pour in egg batter.
When egg begins to thicken, add ratatouille and one or two pieces of goat cheese; fold over onto a warmed plate. Cover to keep warm while repeating with remaining egg batter, ratatouille, and cheese for other omelets.