To Make the Pierogi Dough:

1. Whisk 1 egg, sour cream or yogurt, butter, and salt in a bowl.
2. Add the flours and caraway seeds to a large bowl.
3. Gently stir the wet ingredients into the flour. The dough will initially be very dry and shaggy, seeming as if it will never come together, but have no fear: Keep stirring, and it will pull itself into shape.
4. Once the dough starts to come together, press and smash it against the sides of the bowl with your palms, picking up dough bits and essentially kneading it within the bowl until it forms a ball.
5. Tip the dough and any remaining shaggy flakes out onto a clean work surface or Roul’Pat. Knead until smooth, about 1 minute. Cover the dough with the bowl and let rest 15 minutes.

To Make the Pierogi Filling:

1. While the dough rests, mix the sauerkraut, cheese, and pastrami in a bowl.

To Make the Thousand Island Dressing:

1. Stir the ingredients in a small bowl. (Dressing can be made up to 1 week ahead; cover and refrigerate.)

Assemble the Pierogies:

1. Line a rimmed baking sheet with waxed paper or parchment paper. Have the egg wash at the ready.
2. Divide the rested dough into 4 equal pieces with a bench scraper or knife. Set aside 3 dough pieces and cover with the mixing bowl. Roll remaining dough as thinly as possible into a rough 8- x 12-inch rectangle.
3. Using a 3-inch round cookie cutter, cut out 6 rounds of dough. If the dough isn’t quartered evenly, you may get 5 rounds from one piece and 7 from another. Resist the temptation to re-roll dough scraps for additional rounds. It seems wasteful, but the dough won’t be as tender the second time around.
4. Spoon 1 teaspoon of filling into the center of each dough round. Using your finger, swipe a very scant amount of egg wash—just a light touch—around the dough edge.
5. Fold into a half-moon shape: Either fold the dough over the filling on the work surface—I call this “the blanket”—or gently cup the pierogi in your hand in a U shape—I call this “the taco.” Gently but firmly seal the pierogi by pinching and squeezing the edges together with your thumb and pointer finger. Start with one pinch at the top, then move to one “corner” of the pierogi and pinch along the edge back to the top. Repeat on the opposite side to finish sealing the pierogi.
6. Transfer to the baking sheet and repeat with remaining dough rounds and filling. Freeze on the baking sheet, refrigerate up to 3 hours, or cook immediately.

TO BOIL FRESH OR FROZEN PIEROGIES

1. Boil a pot of water over medium-high heat (fill approximately 1 quart water for every 6 pierogies).
2. Add pierogies and cook until floating, 2 to 3 minutes for fresh and 4 to 5 minutes for frozen.

TO PAN-FRY FRESH OR BOILED PIEROGIES

1. Heat 1 tablespoon neutral oil (like canola or vegetable) or melt 1 tablespoon unsalted butter in a skillet over medium heat.
2. Add as many pierogies as will fit in a single layer without crowding.
3. Cook until pierogies are brown and crispy, about 2 minutes per side. Repeat with additional oil or butter and pierogies.

TO DEEP-FRY FRESH OR FROZEN PIEROGIES

1. Use an electric deep fryer or a large, high-sided pot filled with at least 2 inches of vegetable or canola oil (fill the pot no more than 1/3 full). Heat oil to 350 degrees.
2. Add pierogies and cook until golden brown; frying time varies based on equipment, about 3 minutes for fresh and 5 minutes for frozen.

Line a baking sheet with paper towels. Transfer pierogies to the baking sheet and cool for 1 minute.