Beef Wellington with Mixed Mushrooms

Name: Beef Wellington with Mixed Mushrooms
Ingredients: 2 pounds mixed mushrooms (such as cremini, button, shiitake and/or oyster), trimmed and chopped or torn; 4 tablespoons unsalted butter; 2 shallots, finely chopped; 2 teaspoons fresh thyme, chopped; 1/4 cup cognac or brandy; Kosher salt and freshly ground pepper; 1 center-cut beef tenderloin (2 1/4 to 2 1/2 pounds), trimmed and tied in 1-inch intervals; 1 tablespoon vegetable oil; 4 ounces thinly sliced prosciutto; 2 tablespoons Dijon mustard; 1 large or 2 small (14–17 ounces total) sheets frozen puff pastry, thawed; All-purpose flour, for dusting; 1 large egg, lightly beaten; Flaky sea salt; Chopped fresh chives, for topping