French Onion Chicken Pot Pie

Name: French Onion Chicken Pot Pie
Ingredients: 6 (about 2 1/2 pounds total) bone-in, skin-on chicken thighs; 2 tablespoons, plus 2 teaspoons olive oil, divided; 2 teaspoons kosher salt, divided; 1 teaspoon black pepper, divided; 1 frozen puff pastry sheet, thawed (from 1 [17.3-ounce] package); 1 large egg, beaten; 2 tablespoons unsalted butter; 2 medium-size (about 1 pound, 4 ounces total) yellow onions, halved lengthwise and sliced 1/4-inch-thick (4 1/2 cups); 2 large (6 ounces total) carrots, peeled and cut into 1/2-inch cubes (1 cup); ¼ cup water; 1 tablespoon minced garlic (from 3 garlic cloves); ¼ cup dry sherry; 3 tablespoons all-purpose flour, plus more for work surface; 2 cups beef stock; 2 teaspoons chopped fresh thyme, plus more for garnish; ¾ teaspoon chopped fresh rosemary; 4 ounces shredded Gruyère cheese, shredded (1 cup)