Lightly coat a 3-qt. rectangular baking dish with cooking spray. Spread half of the bread cubes in prepared dish.
In a large skillet heat 2 tsp. of the oil over medium-high heat. Add onion, sweet pepper, and garlic; cook until tender, stirring occasionally. Remove from skillet. Add remaining 2 tsp. oil to skillet. Add mushrooms; cook and stir until tender. Add spinach; cook until slightly wilted. Stir in onion mixture.
In a small bowl stir together both cheeses; reserve 1/3 cup of the cheese mixture. Spread half of the vegetable mixture over bread cubes in dish. Sprinkle with remaining cheese mixture. Layer with remaining bread cubes and vegetable mixture.
In a large bowl whisk together remaining ingredients. Slowly pour over layered mixture in dish. Cover and chill overnight.
Preheat oven to 325 degrees F. Bake, uncovered, 45 minutes; sprinkle with reserved 1/3 cup cheese mixture. Bake 5 to 10 minutes more or until a knife comes out clean. Let stand 10 minutes before serving.