Preheat the oven to 375°F.
Place the cheese and honey in a food processor; whirl until blended, scraping down the inside of the bowl as needed.
Combine the cornstarch and sugar in a medium bowl. Whisk together the egg and 2 teaspoons of the water in another bowl until blended. Set the bowls aside.
Roll the puff pastry into a 12-inch square on a lightly floured work surface. Cut into 9 squares, each 4 inches. Transfer to a baking sheet lined with parchment paper. Gently spread about 11/2 tablespoons of the cheese mixture in a 3-inch-long diagonal on each square.
Gently stir the berries and the remaining 1 teaspoon water into the cornstarch mixture. Continue stirring gently until the dry cornstarch disappears, 2 to 3 minutes. Spoon the mixture over the cheese.
For each Danish, pull a corner over the filling almost to the other side of it and brush the top of the dough with the egg wash. Fold the opposite corner over the first one and press to seal. Leave the remaining corners flat. Brush the dough all over with the egg wash.
Bake the Danishes at 375°F until deep golden and crisp, 30 to 35 minutes. Transfer to cooling racks and serve warm or cool.
Recipes excerpted with permission from One Sheet Eats by Oxmoor House (Time Inc. Books, 2017)