In a large skillet, heat 2 tablespoons olive oil over medium heat. Add prosciutto; cook for 8 to 10 minutes, stirring frequently, until browned and crispy. Remove prosciutto from skillet, and set aside, reserving drippings in skillet. Return skillet to medium heat. Add remaining 1 tablespoon olive oil, mushrooms, onion, garlic, 1/2 teaspoon salt, and black pepper. Cook for 20 minutes, stirring occasionally, until lightly browned. Stir in prosciutto; set aside.
In a medium saucepan, melt butter over medium heat. Add flour; cook for 2 minutes, whisking constantly. Gradually add milk, whisking until smooth. Cook for 5 minutes, whisking constantly. Remove from heat; add cheese, remaining 1/2 teaspoon salt, and white pepper, stirring until cheese is melted.
Preheat oven to 400°. Line 2 rimmed baking sheets with aluminum foil. Brush foil with melted butter.
Cut French bread into 18 (1/2-inch-thick) slices. Spread Dijon mustard onto bread slices. Top with turkey, mushroom mixture, and cheese sauce. Bake for 5 minutes.
Broil for 3 to 5 minutes, or until topping is lightly browned. Garnish with paprika and chopped fresh parsley, if desired.