Set an oven rack in the lowest position and preheat oven to 400°.
Put the puff pastry on a large sheet of parchment. Cover with plastic wrap and roll into a 19-by-7-inch rectangle. Slide pastry onto a half-sheet pan and freeze until firm.
Meanwhile, melt 1 tablespoon butter in a medium skillet over medium heat. Add the beet and 1 tablespoon honey and reduce the heat to low. Cook, stirring gently to coat evenly, until barely tender, about 5 minutes. Add the pear and gently turn to evenly coat. Remove from heat and cool completely.
Put the remaining teaspoon butter in a microwave-safe bowl and microwave for 20 seconds or until melted. Stir in the remaining teaspoon honey, then stir in the sweet potato until evenly coated; set aside.
Pulse the walnuts in a food processor until coarsely chopped. Add the dried plums and pulse until the mixture is finely chopped.
Remove the pastry from the freezer. Using the tip of a sharp paring knife, score a 1-inch border around the edges without cutting through the pastry. Evenly spread the walnut mixture inside the border, then top with the beet, pear, and sweet potato, overlapping them in rows. Reserve any remaining honey mixture in the skillet. If the pastry has softened, refrigerate until firm again.
Bake for 30 minutes, then reduce heat to 350° and continue to bake until pastry is golden brown and vegetables are tender, 10 to 15 minutes more.
Cool tart on the pan on a rack. Right before serving, drizzle with any remaining honey mixture. Drizzle more honey on top if needed to lightly coat the toppings. Scatter with tarragon and serve.