Mushroom Duxelle

• 2 pounds mixed fresh mushrooms (such as oyster, cremini, and shiitakes)
• 7 tablespoons olive oil, divided
• 1 1/4 teaspoons kosher salt, divided
• 1/2 teaspoon black pepper, divided
• 1 large leek (white and light green parts only), halved lengthwise and thinly sliced (2 cups)
• 3 medium garlic cloves, finely chopped
• 2 tablespoons chunjang (Korean black bean paste)
• 2 tablespoons oyster sauce or soy sauce
• 1 tablespoon light brown sugar
• 1/3 cup dry sherry
• 1 cup loosely packed chopped fresh flat-leaf parsley
• 1/2 cup chopped toasted walnuts
• 1/3 cup panko

Cabbage

• 1/4 cup unsalted butter
• 1 small yellow onion, thinly sliced (about 1 cup)
• 5 cups thinly sliced red cabbage
• 1 teaspoon kosher salt
• 3/4 teaspoon black pepper
• Water, as needed
• 5 garlic cloves, finely chopped
• 1 tablespoon fresh thyme leaves
• 1 tablespoon light brown sugar
• 2 tablespoons panko

Additional Ingredients

• 1 (14-ounce) package all-butter frozen puff pastry sheet (such as Dufour), thawed according to package directions
• All-purpose flour
• 1 large egg, beaten