Heat oven to 400°F. Line a baking sheet with parchment paper.
Lay the pie crust of puff pastry on a clean, flat surface that has been very lightly dusted with flour to prevent sticking. Using a clean straight-edge and a pizza roller, cut the pie crust or puff pastry into ½-inch wide strips. Carefully weave the strips into a lattice pattern.
Once the pie crust or puff pastry is woven into a lattice, use a round cookie cutter or the top of a glass to cut out 8 circles. Transfer the circles to a baking sheet with a flat spatula. You may have to pinch pieces of the lattice together at the edges to help keep their shape.
In a small bowl, beat the egg until the yolk and whites are combined. Brush the pastry rounds with beaten egg, then sprinkle each pastry round with sugar. Bake the pastry rounds for 15 minutes. If they are not golden brown, let them bake for an additional 3 minutes. Continue to check and bake until the pastries are golden brown and the sugar starts to caramelize.
Remove the pastry rounds from the oven and let cool completely.
Once the lattice pie crust "cookies" are cooled, flip half of them over. Place a small scoop of ice cream on these flipped-over cookies. Top each of the ice cream scoops with the other cookies and press together. Some ice cream might ooze out through the lattice. OH WELL.
Serve immediately.
You can also assemble the sandwiches in advance, wrap them in plastic, and keep them in the freezer until serving time. I have not tested this beyond a few hours. The pie crusts get a little hard.