Place the cottage cheese in a strainer lined with cheesecloth or heavy-duty paper towels. Pat dry with paper towels and set aside for 5 to 10 minutes.
Place apples in a bowl, and sprinkle with lemon juice to prevent browning.
Combine the cream cheese, drained cottage cheese, 1/2 cup sugar, egg, and vanilla extract in the bowl of an electric mixer, and mix 1 to 2 minutes on high speed, until almost smooth.
Preheat an oven to 400 degrees F (200 degrees C). Line two baking sheets with parchment paper.
Cut each of the puff pastry sheets into 9 pieces; you should have 18 total. Roll the pieces out, one at a time, to make 5 x 5-inch squares. To ensure the pastries puff up, avoid using too much pressure or rolling over the edges of the dough.
Place a square in front of you so that it looks like a diamond. Spoon a generous amount of cheese filling down the middle of each square, leaving 1/4 inch of room at the top and bottom edges. Fan out 4-5 slices of apple down the center of each cheese-filled square. Bring the left and right corners together to meet in the center and pinch to seal.
Transfer the Danish to a baking sheet and repeat with the remaining pastries. Sprinkle sugar over each bundle and bake in the preheated oven until the pastry is golden brown, about 20 minutes.