1. Preheat oven to 350°.
2. Place bread cubes on a jelly-roll pan; coat with cooking spray. Bake at 350° for 20 minutes or until lightly toasted, turning twice. Remove from oven; cool.
3. Heat oil in a large nonstick skillet over medium-high heat. Add shallots and mushrooms; sauté 12 minutes or until lightly browned and moisture evaporates. Remove from heat; stir in parsley, thyme, salt, and pepper.
4. Place half of bread cubes in bottom of an 11 x 7–inch baking dish coated with cooking spray. Arrange mushroom mixture evenly over bread cubes; sprinkle with 1/2 cup fontina and 1 tablespoon Parmesan. Top with remaining bread cubes. Combine milk, broth, and eggs, stirring with a whisk; pour over bread mixture. Gently press with back of a spoon; let stand 30 minutes. Top with remaining 1/2 cup fontina and remaining 1 tablespoon Parmesan.
5. Bake at 350° for 45 minutes or until set. Let stand for 10 minutes before serving. Cut into 6 squares.