Preheat the broiler in your oven. Liberally season chicken with salt and pepper. Lightly coat chicken in flour. In a large sauté pan over medium heat, sear chicken breasts in olive oil until browned and cooked through, about 3 minutes per side. Remove chicken breasts.
Top each piece of chicken with a slice of prosciutto and mozzarella cheese. Place under broiler until cheese is bubbly and melted, 5-7 minutes.
In the same pan used to sear the chicken, melt butter and more olive oil. Sauté garlic and shallots until golden, 2 minutes. Add mushrooms and cook until they are soft and browned, 5 minutes.
Add wine and chicken stock and scrape up the bits stuck to the bottom of the pan. Cook the wine until the sauce is starting to thicken slightly, 10 minutes.
Stir in cream and fresh thyme and cook another 2 minutes. Season with salt and pepper to taste. Use the sauce to top chicken saltimbocca and serve with pasta, gnocchi, or your starch of choice.