Preheat oven to 425°F. Roll out puff pastry on a lightly floured surface into a long rectangle. Cut in half crosswise (halves should be almost square). Transfer to a parchment-lined rimmed baking sheet; prick with a fork in several places. Using a small offset spatula, spread 2 Tbsp. peanut butter in the center of each half to make a 5" round, then spread 2 Tbsp. apple butter over top of each nut-butter round. Pile up apples in the center of each puff pastry and top with peanuts. Dot fruit with butter and sprinkle with sugar. Beat egg and 1 tsp. water in a small bowl, then brush pastry with egg wash and fold up edges around apples, leaving the center open. Press along the folded edges to lightly seal. Brush outside of pastry with egg wash and sprinkle with more sugar. Let chill in freezer 10 minutes. Bake tarts until puff pastry is golden, 15–20 minutes. Reduce oven temperature to 350°F and continue to bake until pastry is deep golden brown and apples are tender, 20–25 minutes. Tarts can be baked 6 hours ahead. Store uncovered at room temperature.