Spoon cottage cheese into a cheesecloth-lined sieve over a medium bowl; let stand 1 hour. Scrape drained cottage cheese into a food processor or blender. Combine sugar and flour. Add sugar mixture to cottage cheese; process until well blended. Scrape seeds from vanilla bean; add seeds to cheese mixture, reserving bean for another use. Add cream cheese to cheese mixture; pulse until smooth. Place cheese mixture in a bowl; chill in freezer for 30 minutes.
Divide Pierogi Dough into 16 equal portions, shaping each into a ball. Working with one portion at a time (cover remaining dough to prevent drying), roll each ball into a 3 1/2-inch circle on a lightly floured surface.
Spoon 1 rounded teaspoon cheese mixture onto half of each circle. Bring 2 opposite sides of circle to center, and pinch to seal, beginning with center and pinching down both sides to form a half-moon shape. Repeat with remaining dough and cheese mixture.
Bring water to a boil in a large saucepan. Add half of pierogi; cook 7 minutes or until done (pierogi will rise to the surface). Remove pierogi with a slotted spoon, and drain in a colander (pierogi will stick to a paper towel). Place pierogi in a single layer on a baking sheet or platter. Repeat procedure with remaining pierogi.
Heat 1 1/2 teaspoons butter in a large nonstick skillet over medium-high heat. Add 8 pierogi; cook 2 minutes on each side or until golden brown. Remove pierogi; keep warm. Repeat procedure with remaining 1 1/2 teaspoons butter and 8 pierogi. Serve with applesauce.