This casserole admittedly isn't the prettiest, but it’s so delicious and it holds a special place in my heart. My mom has been making this for Christmas morning my whole life. Everything you want for breakfast is in this dish: It starts with an egg bread pudding, then gets topped with breakfast sausage, cheese and a homemade mushroom gravy. What makes it extra special is the secret ingredient, dry vermouth. Just a touch goes a long way, adding a fruity yet slightly bitter taste to the top of the casserole that compliments the mushroom gravy. And don't worry, all the alcohol will cook out while it bakes. It's also the perfect make-ahead, and can easily be doubled and baked in a 9-by-13-inch pan to feed a crowd; just remember to adjust the baking time.
To make the mushroom gravy, melt the butter in a large skillet over medium heat. Add the mushrooms and onion and cook until they begin to brown and all the moisture from the mushrooms has evaporated, 8 to 10 minutes.
Add the garlic and cook an additional 1 minute. Stir in the flour until the mushrooms are completely coated and you no longer see any raw white flour, about 1 minute. Slowly pour in the broth and heavy cream, stirring constantly. Bring the mixture to a boil, reduce to a simmer and cook until very thick, 4 to 5 minutes. Season to taste with salt and pepper and let cool. When completely cooled, transfer to an airtight container and refrigerate up to 3 days.
To make the casserole, heat a large skillet over medium-high heat. Add the sausage and cook, breaking up chunks, until browned and cooked through, about 8 minutes. Transfer to a sheet pan lined with paper towels to drain.
Place the bread in an 8-by-8-inch baking dish. Scatter the cooked sausage and then the Cheddar evenly over the bread. Set aside.
Whisk together the eggs, milk, mustard, 1/4 teaspoon salt and pepper to taste in a medium bowl. Pour the egg mixture over the bread, making sure the bread is evenly coated; the egg should come about halfway up the sides of the bread. At this point, cover the casserole dish and refrigerate overnight.
When ready to bake, preheat the oven to 325 degrees F. Remove the casserole from the refrigerator, uncover it and let it sit at room temperature for about 30 minutes while the oven preheats.
Evenly spread the mushroom gravy over the top of the casserole (I like to dollop it on and then spread it with a rubber spatula). Sprinkle the vermouth on top. Place the casserole on a sheet pan and bake until the sides brown and bubble and the egg is set in the center, about 1 hour.
Remove the casserole from the oven and let cool 10 to 15 minutes. Cut into squares and serve immediately. Cool any leftovers completely, then refrigerate; individual servings reheat well in the microwave on the reheat setting.