Preheat oven to 375°. Rub a rimmed baking sheet with 1 tbsp. oil. Spread pumpkin on sheet and roast, uncovered, turning once, till tender when pierced, 30 minutes.
Meanwhile, heat remaining 1 tbsp. oil in a large frying pan over medium heat. Add onion and cook, stirring often, until translucent, about 2 minutes. Stir in spices and cook until fragrant, about 2 minutes.
Transfer onion to a medium mixing bowl. Drain currants and stir into bowl. Add warm pumpkin and mix well, mashing with back of a spoon to create a slightly chunky mixture.
Working with one pie-dough disk at a time, roll dough on a lightly floured surface into a 1/8-in.-thick round. Using a 4-in. ring cutter, cut out 20 circles, rerolling scraps.
Mix 1 tbsp. water into egg and brush onto dough rounds. Spoon 1 tbsp. pumpkin mixture into center of each round, fold over, and seal edges with tines of a fork.
Line 2 baking sheets with parchment. Arrange empanadas on sheets, spaced slightly apart.
Bake until golden brown, 15 to 20 minutes. Serve with chutney.
Make ahead: Follow recipe through step Freeze in an airtight container. Then bake from frozen, 25 to 30 minutes at 375°.
Note: Nutritional analysis is per empanada.