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.
Heat oil in a large nonstick skillet over medium-high heat.
Add shallots and mushrooms; saut 12 minutes or until lightly browned and moisture evaporates.
Remove from heat; stir in parsley, thyme, salt, and pepper.
Place half of bread cubes in bottom of an 11 x 7inch 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.