The strata needs to be refrigerated for at least 4 hours or overnight before baking, so plan accordingly.
Preheat the oven to 375°. Spread the bread on 2 large baking sheets and bake in the upper and lower thirds of the oven for 10 minutes, or until golden and crisp; shift the pans from top to bottom and front to back halfway through.
In a large enameled cast-iron casserole, melt the butter in the olive oil. Add the onion, celery and jalapeño and cook over moderate heat, stirring, until softened, 6 minutes. Add the Italian sausages and cook, breaking up any large pieces, until no longer pink, 8 minutes. Stir in the ham, andouille, thyme and salt and pepper. Transfer to a large bowl to cool.
Add the Gruyère, Parmesan and bread to the meat. In another bowl, whisk the eggs, stock and half-and-half and add to the meat and bread mixture; toss until evenly mixed and moistened. Cover the strata with plastic and refrigerate for at least 4 hours or overnight.
Preheat the oven to 350°. Butter a 4-quart glass or ceramic baking dish. Transfer the strata to the prepared baking dish and smooth the surface. Butter a large sheet of foil and cover the baking dish with it. Bake the strata in the center of the oven for 30 minutes, or until barely set. Remove the foil and bake for 45 minutes longer, or until the strata is bubbling and the top is golden and crusty. Let cool for 15 minutes before serving
Make Ahead: The baked strata can be kept at room temperature for up to 4 hours. Reheat before serving.