French Onion Soup Tarte Tatin Recipe

Name: French Onion Soup Tarte Tatin Recipe
Ingredients: 1/2 batch Old-Fashioned Flaky Pie Dough or 1 sheet frozen puff pastry, thawed; 3 medium yellow onions (about 1 1/2 pounds; 680g) (see note); 4 tablespoons (2 ounces; 56g) unsalted butter, softened, divided; Kosher salt and freshly ground black pepper; 1 tablespoon (15ml) dry sherry (such as Amontillado); 3/4 cup (180ml) homemade chicken or beef stock, or store-bought, low-sodium chicken stock; 1/4 teaspoon Asian fish sauce (optional); 1/4 teaspoon cider vinegar; 1/2 teaspoon (2g) chopped fresh thyme leaves; 1 tablespoon (15g) sugar; 2 ounces (56g) Gruyère cheese, grated; 2 tablespoons (30ml) Dijon mustard (optional); Freshly sliced chives, for garnish