1. To make red-cabbage slaw, whisk together vinegar, sugar, and caraway seeds in a medium bowl. Toss in cabbage. Season with salt and pepper. Let marinate for at least i hour, or up to overnight.
2. To make pierogi dough, whisk together egg, milk, sour cream, and water. Stir in flour a little at a time until dough comes together. Turn out onto a lightly floured board and knead until smooth and elastic, up to 10 minutes. Incorporate more flour if dough is too sticky. Cover dough with plastic wrap and allow to rest for 1 hour.
3. To make filling, place parsnips in a medium saucepan and cover with cold, salted water. Bring to a boil and simmer until tender, 20 to 25 minutes. Drain and put through a food mill to puree.
4. In a small saute pan, cook shallots in butter until soft, 1 to 2 minutes. Stir into puree and add nutmeg, salt, and pepper to taste. Remove from heat and let cool slightly then mix in horseradish and cheese.
5. On a lightly floured board, roll out dough to 1/8 inch. Cut out circles using a 3/4-inch-round cutter. Set circles aside on a floured tray. Place a round T of filling on each circle. Lightly wet edges, fold over, and seal by pinching.
6. To make sauteed apples, heat a medium saute pan over medium-high heat. Melt butter, add apple slices, and toss to coat. Add sugar and toss again to coat evenly. Cook until brown, about 5 minutes.
7. When ready to serve, bring a pot of salted water to a boil. Add pierogies and boil for 5 minutes after they float to the surface; drain. Pierogies can be eaten right away or browned in a small amount of butter. Serve with red-cabbage slaw and sauteed apples.