Preheat oven to 350°F. Coat a 9-inch deep-dish pie pan (6-cup capacity) with cooking spray. Add breadcrumbs, tilting to coat bottom and sides.
Whisk eggs, milk, hot sauce, salt and pepper in a large bowl. Add bread and stir to coat. Set aside in the refrigerator.
Steam broccoli until just tender, 3 to 4 minutes. Refresh under cold water and drain well. Chop coarsely.
Heat oil in a medium nonstick skillet over medium-high heat. Add onion; cook, stirring often, until softened and light golden, 3 to 5 minutes. Add onion mixture and broccoli to the egg mixture; stir in cheese. Pour into the prepared pan, spreading evenly.
Bake the pie until light golden and set, 45 to 50 minutes.
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