Blanch the broccoli in boiling water for 2 minutes, drain and spread it on a towel until completely cool and dry.
Butter a 9-by-13-inch dish. Layer the bagel cubes, broccoli and 2 cups cheddar, then sprinkle the remaining cheese over the top.
Whisk the eggs, milk, salt, dry mustard, pepper and garlic powder. Pour it evenly over the layers and press the bagels gently into the liquid.
Cover and refrigerate for 8 to 16 hours. Let the dish stand out for 30 minutes while the oven heats to 350°F.
Bake uncovered for 55 to 60 minutes, until the center is 160°F and no wet custard remains. Rest 10 minutes before cutting.