Heat the oven to 325°F. Brush the sides and bottom of a 9-by-10-inch baking dish or cake pan (2 inches deep), with the softened butter.
In a small mixing bowl, whisk together the milk and cream.
In a large mixing bowl, whisk the eggs, salt, and pepper with 1 cup of the milk-cream mixture. Add the remaining milk mixture, whisk thoroughly and set aside.
Starting at one end of the loaf, cut the bread into thin, even slices approximately ¼ inch thick. Start assembly of the strata by placing bread slices on the bottom of the dish until the bottom is completely covered. Sprinkle 1 cup of the shredded Asiago cheese on the bread slices to cover them completely. Continue to build the strata with three more layers of bread and two layers of cheese, ending with a layer of bread on top. Gently press the top down. Ladle the custard mixture over the top of the strata.
Cover the strata and let stand at room temperature for 30 minutes, allowing the bread slices to soak in the custard mixture. You can also refrigerate overnight before baking.
Set the strata in the center of the oven and bake for approximately 55 to 60 minutes, until the custard is a rich, dark color and has risen slightly from the pan (it will settle after removal from the oven). The strata should be slightly jiggly in the center. Set it aside to cool for 15 minutes before unmolding.
Run a knife around the outer edges of the pan to loosen. Serve right from the pan, or carefully invert the strata: Place a 10-inch plate over the top of the pan and slowly flip the plate and pan over together, and place the pair on the counter, allowing the strata to fall from the pan onto the plate. If it does not release, cut around the sides of the pan and try again. Serve hot or warm.
Tip: Build the strata inside a soufflé dish and flip it upside down when it's done for a gorgeous presentation at your next brunch.