Preheat oven to 400°F (204°C) with a rack in the center. Line one or two half-sheet pans with parchment, depending on whether you want to bake in batches or one pan at a time.
Place the strawberries, sugar, cardamom, and salt into a small pot and set over medium-high heat.
Bring to a boil, then turn down the heat and simmer, stirring occasionally, for 5 to 10 minutes. The berries will start to soften and break down — you can help them along with the back of a fork or a potato masher if you prefer a smoother filling. Continue cooking until the juices reduce to about 3 tablespoons.
While the fruit cooks, stir together the water, lemon juice, and cornstarch until perfectly smooth.
Stir the cornstarch mixture into the fruit and cook for two minutes, until thickened. Let the mixture cool before proceeding.
Dust a work surface with flour and unfold puff pastry. Dust pastry with flour too. Roll out the puff pastry into a 14-inch square (see note 2) and cut into 16 smaller squares. Place 8 squares onto each parchment-lined baking sheet, leaving a little bit of space in between. (If baking on one pan in two batches, chill the remaining puff pastry until ready to use.)
In a small bowl, whisk together the egg and milk with a fork to make an egg wash.
Place 1 to 2 teaspoons of the cooled strawberry mixture into the center of each pastry square.
Using a pastry brush or your finger, paint the edges of each square with egg wash. Fold each square diagonally into a triangle.
Press the edges together with your fingers to seal, then crimp with a fork.
Paint the pastry with egg wash. Then, using the tip of a paring knife, cut two or three small slits in the top of each triangle to allow the pastries to vent as they bake.
Bake for 20 minutes, until puffed and golden. Transfer to a cooling rack for a few minutes before serving warm or at room temperature.