Lightly butter an 8x8x2-inch baking dish. Set aside.
In a heavy saucepan, heat oil over medium-high heat. Add the shiitake and portobello mushroom slices, garlic, and sage. If youre using dried basil and thyme, add them now. Cook over medium-high heat for about 15 minutes or until mushrooms are tender and browned, stirring occasionally. Remove from heat. If youre using fresh basil and thyme, add them now with the snipped fresh parsley.
In a large bowl, whisk eggs, cream, milk, the 1/4 cup Parmesan cheese, salt, and pepper until well combined. Add bread pieces; toss to coat well. Let stand for 15 minutes.
Stir mushroom mixture into bread mixture. Transfer to the prepared baking dish. Sprinkle with the 1 tablespoon Parmesan cheese.
Bake in a 350 degree F oven for about 1 hour or until pudding is browned and puffed and center is set. Serve warm.