Preheat the oven to 350°F. Grease two 24 cup mini muffin pans. If you only have one pan, bake in two batches and keep the filling and the second batch of rounds in the refrigerator while the first batch bakes.
Unroll the refrigerated crusts on a lightly floured surface (roll homemade dough about 1/8 inch thick). Cut 2.5 inch rounds, about 12 from each crust, 48 in total, rerolling the scraps once. A 3 inch cutter gives a taller rim but only about 9 rounds per crust.
Press each round into a mini muffin cup, easing the dough up the sides so it forms a little rim. Brush the top edges with the beaten egg.
With an electric mixer, beat the room temperature cream cheese and sugar until completely smooth, with no lumps. Add the 2 eggs one at a time, beating after each until combined.
Add the pumpkin puree and beat until smooth, then stir in the vanilla and pumpkin pie spice.
Spoon about 1 tablespoon of filling into each crust, almost to the top of the dough.
Bake for 15 to 20 minutes, until the crust edges are golden and the center of the filling no longer looks wet. The bites puff up in the oven and settle back down as they cool.
Cool in the pan for 5 minutes, then move to a rack to cool completely, or chill for 30 minutes.
Just before serving, top with whipped cream and a pinch of pumpkin pie spice or cinnamon. Keep refrigerated, and do not leave topped bites at room temperature for more than 2 hours.