1. In a medium bowl, cover the currants with hot water and let stand until softened, 20 minutes. Meanwhile, in a small bowl, stir the yeast with 2 tablespoons of warm water and a pinch of sugar and let stand until foamy, 5 minutes.
2. In the bowl of a standing mixer fitted with the dough hook, combine the flour, nutmeg and cinnamon with 1/4 cup of sugar. Add the milk, egg, egg yolk and half of the softened butter; beat at low speed for 3 minutes. Beat in the yeast, then add the salt. Beat the dough on medium speed until soft and silky, about 8 minutes; the dough should pull cleanly away from the bowl.
3. With the machine on, add the remaining softened butter to the dough in walnut-size lumps, beating at low speed between additions until incorporated. Drain the currants, pressing out any excess water; beat them into the dough on low speed. Transfer the dough to a greased bowl, cover and let stand in a warm, draft-free spot until doubled in bulk, 1 hour. Punch it down, reform into a ball and return to the bowl. Cover and let stand until billowy, 1 hour.
4. Butter 2 large baking sheets. Turn the dough out onto a work surface and cut it into 12 equal pieces. Pinch each piece into a ball and arrange 6 balls on each of the prepared baking sheets, smooth sides up. Cover with plastic wrap and let stand for 10 minutes. Using lightly floured hands, press each ball into a flat 4-inch disk. Using a 1 1/4-inch round cutter, stamp out the centers of each disk and return the holes to the baking sheets. Cover loosely with plastic wrap and let stand for 1 hour, until risen slightly.
5. Position racks in the upper and lower thirds of the oven and preheat the oven to 400°. Bake the doughnuts and holes for 25 minutes, shifting the pans from top to bottom and front to back halfway through; the doughnuts are done when they are golden and puffy and an instant-read thermometer inserted into the thickest part registers 200°.
6. Spread sugar in a shallow bowl. Brush the hot doughnuts and holes on both sides with the melted butter and dredge them in sugar. Transfer them to a platter and serve.