1. Gather all the ingredients for the dough and filling.
2. Sift 2.5 cups of all-purpose flour with a pinch of salt into a large mixing bowl for the dough.
3. Form a mound with the flour and create a well in the center.
4. Crack one large egg into the well and pour in 0.5 cups of lukewarm water gradually.
5. Knead the dough for 10-15 minutes until it becomes smooth and elastic. Adjust water or flour as needed to get the right consistency.
6. Form the dough into a ball, cover with a damp cloth, and let it rest at room temperature for 30 minutes to 1 hour.
7. While the dough is resting, peel, core, and finely dice or grate 1 1/4 pounds of apples.
8. Cook the apples in a non-stick skillet over medium-low heat for 10 minutes, stirring frequently.
9. Add 7 tablespoons of sugar and 1-2 teaspoons of ground cinnamon to the apples, stirring to combine.
10. Optional: Add 1 tablespoon of lemon juice for a tangy flavor.
11. Stir in 1 tablespoon of unsalted butter and continue cooking for 5-7 minutes until the filling reaches the desired consistency.
12. Set the apple filling aside to cool.
13. Lightly flour a clean surface in preparation for rolling out the dough.
14. Roll out the rested dough to approximately 1/8 inch thickness with a rolling pin.
15. Cut the dough into 2.5-inch circles using a cutter or the rim of a glass.
16. Place about a tablespoon of cooled apple filling in the center of each dough circle.
17. Fold the dough over the filling to create a half-moon shape.
18. Press the edges firmly to seal and use a fork to crimp for a decorative finish.
19. Repeat until all the dough and filling are used, placing prepared pierogi on a floured surface.
20. Boil a large pot of salted water.
21. Gently drop the pierogi into the boiling water in batches.
22. Cook until they float to the surface, then simmer for an additional 2-3 minutes.
23. Remove the pierogi with a slotted spoon and briefly rinse under cold water.
24. Optional: Heat a non-stick skillet with butter or oil over medium heat.
25. Pan-fry boiled pierogi for 2-3 minutes on each side until golden and crispy.
26. Serve the pierogi as desired, dusted with powdered sugar, with sour cream, or drizzled with caramel sauce.
27. Enjoy your homemade apple pierogi.