Preheat the oven to 350°F (175°C). Drizzle sweet potatoes with 1 tablespoon melted butter and sprinkle with 1/4 teaspoon each salt and pepper. Wrap each potato in foil and put on a 10x15-inch rimmed baking sheet.

Bake sweet potatoes in the preheated oven until tender, about 1 hour.

Let sweet potatoes cool completely, about 1 hour. (Alternatively, chilled sweet potatoes can stand at room temperature for 30 minutes before use.)

Meanwhile, combine broth, rice, garlic, and 1/4 teaspoon salt in a medium saucepan. Bring to a boil over medium-high heat, then reduce heat to low. Simmer covered until rice is tender, about 40 to 45 minutes; drain any excess water.

Melt 1/4 cup butter in a large skillet over medium heat. Add mushrooms and onion; cook, stirring occasionally, until liquid evaporates and mushrooms and onion are tender, about 10 to 12 minutes. Add sherry and cook until evaporated, about 2 minutes. Stir in tarragon, remaining salt and pepper, rice mixture, and goat cheese until combined. Let cool completely.

Roll puff pastry sheets into 6.5-inch squares and pile filling onto the center of each square (about 1/3 cup per serving). Top each with a sweet potato.

Beat together egg and water, brush edges of pastry squares, and fold corners over potatoes to seal. Arrange pastry bundles seam side down on a baking sheet. Decorate using remaining pastry and brush with egg wash.