Cheesy Brussels Sprouts Morsel 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
• 1 teaspoon sea salt
• 1 teaspoon black pepper, divided
• 8 ounces fresh shiitake mushrooms, sliced
• 4 tablespoons salted butter, divided
• 10 large fresh sage leaves, finely chopped (about 2 tablespoons)
• 5 large eggs
• 3 cups whole milk
• 4 ounces Gruyère cheese, shredded (about 1 1/4 cups)
• 5 ounces Parmesan cheese, grated (about 1 1/2 cups)