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 1 hour, or overnight.
To make pierogi dough, whisk together egg, milk, sour cream, and 1/2 cup 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.
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.
In a small saute pan, cook shallots in butter until soft, stirring occasionally, 1 to 2 minutes. Stir shallots into puree, and add nutmeg, salt, and pepper to taste. Remove from heat, and let cool slightly, then mix in horseradish and cheese.
On a lightly floured board, roll out dough to 1/8 inch. Cut out circles using a 3 3/4-inch-round cutter. Set circles aside on a floured tray. Place a round tablespoon of filling on each circle. Lightly wet edges, fold over, and seal by pinching.
To make sauteed apples, heat a medium sautee 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.
When ready to serve, bring a pot of salted water to a boil. Add pierogi, and cook for 5 minutes after they float to the surface; drain. Pierogi can be eaten right away or browned in a small amount of butter. Serve with red-cabbage slaw and sauteed apples.