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Boil water in steamer or large saucepan over high heat. Reduce heat to medium-high. Place broccoli in steamer basket; cook for 3 to 4 minutes, or until tender-crisp and bright green. Place in ice water bath to stop cooking process. Drain and coarsely chop. Set aside.
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Preheat oven to 350º F.
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Heat oil in a medium nonstick skillet over medium heat.
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Add onion; cook, stirring frequently, for 4 to 6 minutes, or until onion is translucent.
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Add garlic; cook, stirring frequently, for 1 minute. Remove from heat. Set aside.
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Combine eggs, milk, spinach, salt, pepper, broccoli, and onion mixture in a large bowl; mix well.
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Place bread and cheese in 3-quart baking dish that has been lightly coated with spray.
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Top with egg mixture. Press down gently so that bread absorbs egg mixture. Cover with aluminum foil.
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Bake for 20 minutes. Remove foil; bake for an additional 15 to 25 minutes, or until strata has puffed up, is cooked through, and is lightly browned on top.
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Tip: Strata can be covered and refrigerated overnight before baking. (This may increase baking time.) This works well if you are serving it for brunch.