Preheat oven to 375°.
Heat oil in a large nonstick skillet over medium-high heat. Add mushrooms, onion, and shallots; sauté 5 minutes or until onion and shallots are lightly golden. Stir in 1/4 cup wine, thyme, 1/2 teaspoon salt, and 1/4 teaspoon pepper; cook 1 minute or until liquid has evaporated, scraping pan to loosen browned bits. Set aside.
Melt butter in a large saucepan over medium-high heat. Add flour; cook 30 seconds, stirring constantly with a whisk. Add remaining 1/2 cup wine, broth, and milk; bring to a boil, stirring constantly. Reduce heat, and cook 5 minutes or until slightly thick, stirring constantly. Remove pan from heat, and stir in fontina and 1/4 cup Parmigiano-Reggiano, remaining 1/2 teaspoon salt, and remaining 1/4 teaspoon pepper. Stir in mushroom mixture, pasta, and turkey; toss gently to combine. Place in a 13 x 9-inch baking dish coated with cooking spray; sprinkle with remaining 1/4 cup Parmigiano-Reggiano. Bake at 375° for 12 minutes or until lightly browned. Let stand 10 minutes before serving.