Preheat the oven to 350 degrees F. (Only if you will be baking these right away.) Lightly grease a sheet pan or cover with parchment paper.
Add the spinach to a medium sauté pan over medium-high heat. Heat, stirring frequently, until most of the moisture has been cooked off. Add the chopped artichoke hearts and cook, stirring frequently, just until the artichoke hearts are heated through, about 3 minutes more. Remove the pan from the heat and immediately add the cream cheese, mayonnaise, Parmesan, salt, and pepper. Stir until evenly combined.
Roll puff pastry out on a lightly floured surface until it measures about 15x20 inches. Using a round biscuit cutter or the top of a drinking glass, cut the pastry into even circles. (Alternatively, you can use a pizza cutter to slice the pastry into even squares or rectangles. Any shape will work.) Place a large spoonful of the spinach and artichoke mixture in the center of each piece of dough. With each piece of pastry, fold one side over so that the ends meet and crimp the edges to seal the filling inside. Place the hand pies on the prepared baking sheet, leaving about 3 inches of space between them. Using a sharp knife, cut three slits in the top of each hand pie and lightly brush the top with egg white.
Bake until the tops of the pies are golden brown, 15-20 minutes. Serve immediately.