Make the filling first. Melt 1 tablespoon (14 g) butter in a medium skillet over medium heat. Add the diced apples, brown sugar, 1 teaspoon (2.6 g) cinnamon, lemon juice, cornstarch, and salt. Cook, stirring often, for 6 to 8 minutes, until the apples are tender and the juices are thickened. Transfer to a bowl and cool completely.

Make the dough. In a large bowl, stir together the flour and salt. Add the egg, warm water, and 1 tablespoon (14 g) melted butter. Stir until a shaggy dough forms, then knead on a lightly floured surface for 5 to 7 minutes, until smooth and soft. Cover the dough and let it rest for 20 minutes.

On a lightly floured surface, roll the rested dough to about 1/8 inch (3 mm) thick. Cut out rounds with a 3-inch (7.5-cm) cookie cutter or the rim of a glass. Gather and reroll the scraps as needed.

Place about 1 1/2 teaspoons (7 to 8 g) of cooled apple filling in the center of each dough round. Fold the dough over the filling to form a half-moon. Press the edges firmly together, pinching or crimping them to seal. Keep the shaped pierogi on a lightly floured tray and cover them with a clean towel while you finish the rest.

Bring a large pot of lightly salted water to a gentle boil. Cook the pierogi in batches so they have room to move. Once they float, cook for 2 to 3 minutes more, then lift them out with a slotted spoon and drain well.

In a wide bowl, combine the granulated sugar and remaining 1 teaspoon (2.6 g) cinnamon. Brush or toss the warm drained pierogi with the 3 tablespoons (42 g) melted butter, then coat them in the cinnamon sugar. Serve warm with sour cream, whipped cream, or vanilla ice cream if desired.