1. In a large skillet, cook bacon until crisp. Transfer bacon to plate lined with paper towels to drain, reserving 2 tablespoons drippings in the skillet. Crumble bacon; set aside. Add onion to skillet; cover and cook over medium-low heat for 15 minutes, stirring occasionally. Uncover and cook over medium heat until caramelized, about 5 minutes. Meanwhile, cook broccoli in a small amount of lightly salted water for 3 minutes; drain.
2. In same skillet; cook onion in bacon drippings, covered, over medium-low heat for 15 minutes, stirring occasionally. Uncover; cook over medium heat about 5 minutes, until caramelized. Meanwhile, in a small saucepan cook broccoli in enough lightly salted water to cover for 3 minutes. Drain and set aside.
3. In a large bowl combine eggs, milk, basil, salt and pepper. Stir in bread cubes, broccoli, caramelized onion, cheese, and crumbled bacon. Transfer to a 2-quart baking dish. Cover and refrigerate for 2 to 24 hours.
4. Preheat oven to 325 degrees F. Bake casserole, covered, for 20 minutes. Uncover and bake 20 to 30 minutes more or until a knife inserted near center comes out clean. Let stand 10 minutes before serving.
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