Preheat oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.
Heat vegetable oil in a skillet over medium heat. Add onion and carrot; cook and stir until softened, about 5 minutes.
Transfer onion and carrot to a food processor. Add bread crumbs. Chop meatless patties coarsely and add to the food processor. Pulse until combined, 4 to 5 times. Add egg; pulse several more times.
Place puff pastry on a lightly floured surface; roll out slightly. Add 3 rows of meatless patty mixture across the pastry. Cut into 3 sections using a pizza wheel. Press edges of pastry together and roll over to seal.
Brush milk over tops of pastry and prick with a fork. Sprinkle poppy seeds over milk. Cut rolls into 2-inch pieces using the pizza cutter; arrange on the prepared baking sheet.
Bake in the preheated oven until pastry is puffed up and golden brown, about 20 minutes.