Preheat oven to 375°. In a food processor, whirl cheese and honey until blended, scraping down inside of bowl as needed.
Combine cornstarch and sugar in a medium bowl. In another bowl, whisk egg and 2 tsp. water to blend. Set bowls aside.
Roll puff pastry into a 12-in. square on a lightly floured work surface. Cut into 9 squares, each 4 in. Transfer to a baking sheet lined with parchment paper. Gently spread about 1 1/2 tbsp. cheese mixture in a 3-in.-long diagonal on each square.
Gently stir berries into cornstarch mixture with 1 tsp. water. Continue stirring gently until dry cornstarch disappears, 2 to 3 minutes. Spoon mixture over cheese.
For each danish, pull a corner over filling almost to other side of it and brush top of dough with egg wash. Fold opposite corner over first one and press to seal. Leave remaining corners flat. Brush dough all over with egg wash.
Bake danishes until deep golden and crisp, 30 to 35 minutes. Transfer to cooling racks and serve warm or cool.