Cheesy Brussels Sprouts Bread Pudding

Name: Cheesy Brussels Sprouts Bread Pudding
Ingredients: 2 tablespoons extra-virgin olive oil; 2 large shallots, finely sliced; 12 ounces brussels sprouts, trimmed and thinly sliced (about 4 cups); 2 large garlic cloves, finely chopped; 2 teaspoons kosher salt, divided; 1 teaspoon black pepper, divided; 8 ounces fresh shiitake mushrooms, sliced; 1/4 cup salted butter, divided; 10 large fresh sage leaves, finely chopped (about 2 tablespoons); 5 large eggs; 3 cups whole milk; 1 (14-ounce) loaf brioche bread, cut into 1-inch cubes; 4 ounces Gruyère cheese, shredded (about 1 1/4 cups); 5 ounces Parmesan cheese, grated (about 1 1/2 cups)