Chicken Pot Pie Sandwich

Name: Chicken Pot Pie Sandwich
Ingredients: 2 cups self-rising flour; 1/2 teaspoon baking soda; 1 stick cold salted butter, cubed, plus 4 tablespoons, melted; 1 cup cold buttermilk; 1 1/2 teaspoons chopped fresh thyme; Crushed black pepper; 4 bone-in, skin-on chicken breasts; 2 tablespoons butter; 2 carrots, diced; 1/2 yellow onion, diced; Kosher salt and freshly ground black pepper; 2 tablespoons all-purpose flour; 2 tablespoons dry sherry; 3/4 cup heavy cream; 3/4 cup chicken stock; 1 sprig fresh thyme; 1 cup frozen peas, defrosted; 1/4 cup fresh parsley leaves; 1/4 cup toasted pine nuts; 2 tablespoons lemon juice; 1/4 cup extra-virgin olive oil