1Preheat oven to 350. In bowl, stir together bagel cubes, melted butter, salt and pepper to coat well. Spread cubes on baking sheet. Bake, stirring several times, halfway through til golden, about 25 minutes. Let cool.
2In sauté pan over medium high heat, warm 1 Tbsp. oil. Add half of mushrooms, salt and pepper., Cook, stirring occasionally, til tender, 5 to 6 minutes. Transfer to paper towel lined plate. Repeat with 1 Tbs. oil and remaining mushrooms. In same pan over medium high heat, warm 1 Tbsp. oil. Add asparagus; cook, stirring occasionally until just tender, about 4 minutes. transfer to paper towel lined plate.
3In bowl, stir together 6 cups bagel cubes, mushrooms, asparagus, peas, sausage and 4 oz. cheese. Transfer to 2 quart baking dish. in bowl, beat together eggs and half and half; pour over bagel mixture. Cover; refrigerate at least 1 hour or up to overnight.
Preheat oven to 350. In food processor, pulse remaining bagel cubes into coarse crumbs. Uncover strata, bake 50 minutes. Sprinkle with 2 oz. cheese and bagel crumbs. Bake 10 minutes more. serves 8 to 10