In a small saute pan over medium heat, heat 1 tablespoon olive oil and saute shallot until translucent. Add mushrooms, garlic, carrots and saute for 10 to 15 minutes, until most of the liquid from mushrooms has evaporated. Remove from heat and pour mixture into a large bowl. Let cool. Stir in parsley, bread crumbs, egg, balsamic vinegar and salt and pepper to taste. The mixing spoon should stand in the mixture. If not, add more bread crumbs Preheat oven to 400 degrees F.
Shape mushroom mixture into 4 burgers. Heat remaining oil in a large saute pan over medium high heat and sear until golden on both sides. Crumble cheese over the top and place pan in oven for 15 minutes until lightly browned. Remove each burger to a bun and serve with traditional garnish of sliced onion, tomato and fries.