Make the pierogi dough: Whisk the applesauce, vegetable oil, 1/2 teaspoon salt, 1 egg and 1/2 cup warm water in a bowl. Put the flour in another bowl and stir in the applesauce mixture. Turn the dough out onto a floured surface and knead until smooth, 1 to 2 minutes, dusting with flour as needed. Cover with plastic wrap and set aside, about 30 minutes.
Meanwhile, make the filling: Cook the bacon in a large skillet over medium heat until crisp. Transfer to a paper towellined plate. Drain all but 2 tablespoons fat from the skillet. Add the mushrooms, season with salt and cook over high heat, stirring until browned, about 4 minutes. Add the onion and cook until soft, about 4 more minutes.
Stir in the sauerkraut, thyme, paprika, 1 cup applesauce and 1/2 cup water. Bring to a simmer, cover and cook until the mushrooms are tender, about 5 minutes, then uncover and cook until thickened, about 4 minutes. Chop the bacon and add to the skillet with the remaining 1/4 cup applesauce and salt to taste.
Make the pierogies: Bring a large pot of salted water to boil. Beat the remaining egg in a bowl with 1 tablespoon water. Roll out half of the dough to less than 1/8 inch thick (keep the remaining dough covered with plastic wrap). Cut into rounds using a 3 1/2-inch cookie cutter or a drinking glass. Top each round with 1 tablespoon of the mushroom filling, brush the edges with the beaten egg and fold in half; crimp with a fork to seal. Repeat with the remaining dough and filling.
Melt 4 tablespoons butter in a large skillet over medium-low heat. Boil the pierogies in batches until tender, about 5 minutes, then transfer to the skillet using a slotted spoon and toss with the butter until browned. Serve with applesauce and sour cream.