Preheat the oven to 400 degrees F.
Put the bell pepper, carrots, and squash into a saucepan with a small amount
of boiling water. Cover the pan, and steam for 4 minutes over medium-high heat. Drain, and transfer the vegetables to a bowl, and toss them together with the green onions, corn, cheese, cilantro, and salt, to taste. Set the bowl aside until the mixture has cooled completely.
On a lightly floured surface, with a lightly floured rolling pin, roll the puff pastry sheets out to 1/4-inch thickness. Cut the pastry into 6 to 8, 3 1/2 to 4 1/2-inch rounds (using biscuit or cookie cutters), depending on whether you want large or small patties. Using a fork, prick the dough all over.
If you are preparing 8 patties, spoon 1/8 of the filling towards the side of 1 pastry round, leaving a 1/2-inch barrier around the edge. If you plan on making 6 patties, divide 1/6 of the filling between the larger pastry rounds.
Brush the bare pastry edge around the filling with some of the beaten egg. Fold the other half of the circle over the filling and press the edges together to seal the patty, using a fork for a crimping effect.
Glaze the patties with the remaining egg wash, place them onto a sheet pan lined with parchment paper, and bake in the preheated oven for 25 minutes, or until the patties are nicely golden brown on the outside as well as the inside.