Preheat the oven to 400°F and line a baking sheet with parchment paper to prevent sticking.
In a small bowl, stir together the softened cream cheese, sugar, and vanilla until smooth and well blended. Set aside.
Unfold the puff pastry sheets and cut each one into thirds lengthwise, then cut each strip in half to create 12 rectangles total. Place them on the prepared baking sheet, leaving space between each. Using a butter knife, lightly score a border about ¼-½ inch from the edge of each pastry, being careful not to cut all the way through.
Spoon the cream cheese mixture into the center of each pastry, staying inside the scored border. Top with a small amount of strawberry jam.
In a small bowl, beat the egg and brush it along the edges of each pastry to help them bake up golden.
Bake for 12-14 minutes, or until the pastries are puffed and lightly browned.
While they cool slightly, whisk together the powdered sugar and milk until smooth, then drizzle the glaze over the warm pastries.
Serve and enjoy.