Step 1Heat the oven to 375 degrees and butter a 9-by-13-inch baking dish. In a medium saucepan, combine the milk, garlic, sage, thyme and black pepper. Heat over high heat until just before the milk reaches a simmer. Remove from the heat and allow to cool slightly.
Step 2Heat the butter and oil in a large skillet over medium heat. Add the mushrooms and salt and saute, stirring often, until the mushrooms are soft and fragrant and beginning to caramelize, about 6 minutes. Remove from the heat; set aside.
Step 3In a large bowl, whisk the eggs. Whisk in the cooled milk. Add the bread pieces, cheeses and sauteed mushrooms and stir until well combined. Pour the mixture into the baking dish, pressing down on the bread to make sure it is submerged. Let the mixture sit for about 15 minutes while the bread absorbs the liquid.
Step 4Bake for 35 minutes, until golden brown, rotating once for even cooking. Serve immediately.