In one small bowl, whisk together the softened cream cheese, half the sugar, and the vanilla extract until smooth. In a separate bowl, stir the pumpkin puree together with the remaining sugar and the pumpkin pie spice until fully combined.
Lightly roll out your thawed puff pastry sheet on a floured surface just to smooth any fold lines. Slice it into 12 even rectangles or squares using a sharp knife, keeping the pieces roughly uniform so they bake evenly.
Score a smaller rectangle within each piece of pastry, leaving a half-inch border you won't fill. Spoon a small dollop of the cream cheese mixture into the center, then top it with a bit of the pumpkin filling, gently swirling the two together with a toothpick.
Brush the exposed border of each pastry with the beaten egg for that golden, glossy finish. Bake in a preheated 400°F oven for 12 to 15 minutes, until the edges are puffed and deeply golden brown.
Let the danish cool for about 10 minutes on the baking sheet before glazing, since a hot pastry will melt the glaze right off. Whisk together the powdered sugar and milk, then drizzle it generously over each danish before serving.