Step 1In a large sauté pan, melt the butter and oil over medium-high heat. Add the onion and cook, stirring frequently, until the onion is softened and just beginning to color, 8 to 10 minutes. Stir in the mushrooms and continue to cook, stirring frequently, until the mushrooms soften and begin to give up their moisture. Continue cooking until most of the liquid has evaporated from the mushrooms and both the mushrooms and onions are a rich golden.
Step 2Remove the pan from the heat and add the wine. Return the skillet to the heat and stir, scraping any flavoring from the bottom of the pan, until the wine is absorbed, 2 to 3 minutes. Remove from heat, stir in the thyme and season with one-half teaspoon salt and one-fourth teaspoon of pepper, or to taste. Set aside.
Step 3In a large bowl, whisk together the eggs and egg yolks. Beat in the half-and-half, then stir in the mushroom mixture. Stir in the bread cubes, then the cheese. Place the bread pudding base into a greased 13-by-9-inch baking dish or casserole. Cover loosely and refrigerate the mixture until ready to bake, up to overnight.
Step 4Heat the oven to 375 degrees. Remove the baking dish from the refrigerator to warm slightly as the oven heats.
Step 5Bake the bread pudding, uncovered, until the pudding is well-puffed and the topping is golden, about 45 minutes. Serve immediately, with the prepared sausage gravy on the side.
Step 1In a dry skillet, cook the sausage over medium-high heat until the fat is rendered and the sausage is golden brown, about 10 minutes. Strain the sausage, leaving the fat in the pan. Add the butter and onion, and cook, stirring frequently, until the onion is softened, 6 to 8 minutes.
Step 2Stir in the garlic, black pepper and red pepper flakes. Slowly stir in the flour, mixing well to avoid any lumps. Slowly add the half-and-half, whisking well to avoid lumps. Bring the mixture to a gentle simmer and cook, uncovered, for about 20 minutes to cook out the raw flour taste and thicken the gravy. Stir frequently to prevent the flour from settling and scorching on the bottom of the pan.
Step 3About 5 minutes before the gravy is ready, stir in the sausage and fresh thyme. Taste and adjust the seasoning as desired with salt and pepper. This makes about 4 cups of gravy.