In a bowl, stir together the bagel cubes, melted butter, salt and pepper to coat well. Spread the cubes in a single layer on a baking sheet. Transfer to the oven and bake, stirring the cubes several times halfway through, until golden and toasted, about 25 minutes. Let cool.

In a sauté pan over medium-high heat, warm 1 Tbs. of the oil. Add half of the mushrooms and season with salt and pepper. Cook, stirring occasionally, until browned and tender, 5 to 6 minutes. Transfer to a paper towel-lined plate. Repeat with 1 Tbs. of the oil and the remaining mushrooms.

In the same pan over medium-high heat, warm the remaining 1 Tbs. oil. Add the asparagus and cook, stirring occasionally, until just tender, about 4 minutes. Transfer to a paper towel-lined plate.

In a bowl, stir together 6 cups of the bagel cubes, the mushrooms, asparagus, peas, sausage and 4 oz. of the cheese. Transfer to a 2-quart baking dish. In a bowl, beat together the eggs and half-and-half and pour over the bagel mixture. Cover the dish with plastic wrap and refrigerate for at least 1 hour or up to overnight.

Preheat an oven to 350°F.

Pulse the remaining bagel cubes into coarse crumbs in a food processor. Remove the plastic wrap from the strata, transfer to the oven and bake for 50 minutes. Sprinkle with the remaining 2 oz. cheese and the bagel crumbs, then bake until the cheese is melted, about 10 minutes more. Serves 8 to 10.