Preheat your oven to 400°F (200°C). Line a large baking sheet with parchment paper and set aside.

Heat the olive oil in a medium skillet over medium heat. Add the asparagus, salt, and black pepper, stirring to coat evenly. Cover the skillet and cook for about 5 minutes, stirring occasionally, until the asparagus is tender-crisp. Remove from the heat.

Transfer the cooked asparagus to a mixing bowl. Add the cooked rice and shredded cheddar cheese. Stir until everything is evenly combined. The warmth from the asparagus will begin melting the cheese slightly, helping the filling hold together.

Lightly flour a clean work surface. Roll each puff pastry sheet into an approximately 11-inch square. Cut each sheet into 4 equal squares, giving you a total of 8 pastry squares.

Working with one pastry square at a time, brush a thin border of beaten egg (or milk) around the edges. Place about 1/2 cup of filling in the center of the square. Fold one corner over to meet the opposite corner, creating a triangle. Press the edges together firmly to seal. Use the tines of a fork to crimp the edges and secure the filling inside. Transfer the finished pies to the prepared baking sheet, spacing them about 1 inch apart.

Brush the tops and sides of each pastry with the remaining egg wash or milk. Cut a small slit in the top of each pie to allow steam to escape. Bake for 20 to 25 minutes, or until the pastries are puffed, golden brown, and beautifully flaky.

Allow the puff pies to cool on the baking sheet for about 5 minutes before serving. Enjoy warm while the cheese is still deliciously melty.