Preheat broiler to high with oven rack in top third of oven. Place bread pieces in a food processor; process until finely crumbled, about 30 seconds. Transfer to a small bowl; toss with 1 tablespoon oil, and set aside.
Combine broccoli and 1/4 cup water in a large microwavable glass bowl; cover loosely with plastic wrap. Microwave at high until crisp-tender, about 5 minutes. Drain; set aside.
Heat 1 tablespoon oil in a large skillet over medium-high. Add mushrooms; cook, stirring occasionally, until golden, about 8 minutes. Transfer to a bowl; set aside.
Wipe skillet clean; add remaining 1 tablespoon oil, and heat over medium. Add onion; cook, stirring occasionally, until softened, about 5 minutes. Add garlic; cook, stirring constantly, until fragrant, about 1 minute. Sprinkle with flour; cook, stirring constantly, until onion pieces are evenly coated, about 1 minute. Whisk in milk; bring to a simmer over medium-high, stirring occasionally. Simmer, stirring occasionally, until thickened, about 4 minutes. Whisk in mayonnaise, salt, pepper, and nutmeg until smooth. Add broccoli and mushrooms; stir to combine. Cook, stirring occasionally, until mixture is hot, about 2 minutes.
Transfer broccoli mixture to an 11- x 7-inch glass or ceramic baking dish lightly coated with cooking spray; sprinkle with breadcrumbs and cheese. Broil until golden and bubbling, about 3 minutes.