Make dough: Combine flour, cold water, beaten egg, and salt in a bowl; mix until a smooth dough forms. Cover the bowl and set aside.

Make potato filling: Place potatoes in a large pot. Add water to cover, bring to a boil, and cook until tender, 25 to 35 minutes. Drain and mash in the pot until smooth.

Place bacon in a large skillet and cook over medium-high heat, turning occasionally, until evenly browned, about 10 minutes. Drain bacon slices on paper towels. Crumble and stir into mashed potatoes. Stir in cheese; season with salt and pepper. Set aside.

Make sauerkraut filling: Place bacon in the skillet and cook over medium-high heat, turning occasionally, until evenly browned, about 10 minutes. Drain, crumble, and place in a medium bowl. Stir in sauerkraut, then sour cream. Mix well.

Roll dough out on a floured surface to 1/8- to 1/4-inch thickness. Cut out 3-inch circles with a cookie cutter or an upside-down drinking glass. Place a spoonful of potato or sauerkraut filling in the center of each circle and fold over, pinching edges together to seal.

Bring a large pot of lightly salted water to a boil; drop perogies in boiling water and cook until they float, 4 to 5 minutes.