For toast, broil bread til golden on both sides, about 5 minutes. Let bread cool.
Whisk eggs, cream, and 2 tsp. thyme in shallow dish; season with salt and pepper. Add bread; soak 10 minutes, turning often.
For sauce, sauté mushrooms in oil in sauté pan til browned, 6 to 7 minutes. Transfer mushrooms to bowl. Sauté onion in same pan in 2 Tbsp. butter, 1 to 2 minutes. Stir in tomato paste and flour; cook 1 minute. Whisk in sherry til thickened, 1 minute. Stir in broth and mushrooms; cook 2 minutes. Whisk in cream cheese and 1 tsp. thyme. Keep warm.
Cook bread in 1 Tbsp. butter in cast iron skillet til browned on both sides, about 5 minutes. Serve French toast with mushroom sauce. Serves 4