Preheat oven to 400F. Spray a large baking sheet (or 2 smaller baking sheets) with non-stick spray.
Using a stand-up mixer, whisk together the cream cheese, 1/4 cup sugar, egg yolk (reserve the egg white for step 3!), 1 teaspoon lime juice, and a pinch of vanilla bean powder until smooth and creamy.
In a small bowl, combine the egg white and 1 teaspoon water, and mix.
Dust your working surface with a bit of flour. Carefully unfold one sheet of puff pastry over the flour. (Place the other sheet in the fridge while working, to keep it cold) Cut the puff pastry into 9 pieces. Place about 1 tablespoon of the cream cheese filling in the center of each puff pastry square and press down.
Fold in the corners of each puff pastry to make an octagon. Place 5-6 mulberries in the center of each danish (make sure you save 4-5 berries for step 7 to make the pink glaze!). Brush the exposed puff pastry dough with the egg white wash.
Carefully transfer each danish onto the baking sheet, and bake for 17-21 minutes, or until the puff pastry is golden brown and the cream cheese filling is set.
White the puff pastries are baking: In a small bowl, whisk together the remaining lime juice (almost 2 tablespoons) and 3/4 cup powdered sugar until a smooth glaze forms. Split the glaze into 2 bowls (optional, see notes). Use your fingers to squeeze 3-4 mulberries into one of the bowls and mix well (or squeeze the berries between two spoons to keep your hands cleaner). Drizzle the mulberry danishes with both the pink glaze and the white glaze.
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