Preheat your oven to 200°C (400°F). Line a large, sturdy baking sheet with a fresh piece of parchment paper.

In a medium mixing bowl, vigorously blend the softened cream cheese and extra virgin olive oil until smooth. Fold in the thoroughly dried spinach, diced artichokes, shredded mozzarella, crumbled feta, grated parmesan, minced garlic, red pepper flakes, parsley, oregano, salt, and black pepper. Stir until a uniform mixture forms.

Lightly dust a clean work surface with flour. Unroll the chilled puff pastry sheets. Use a rolling pin to smooth any creases, then cut each sheet into 4 equal squares using a sharp knife, creating 8 squares total.

Arrange 4 pastry squares onto your prepared baking sheet. Divide the spinach artichoke filling evenly among them, mounding it in the center while leaving a clean 0.5-inch border around the edges.

Whisk the egg and water together in a small bowl. Lightly brush this egg wash onto the exposed borders of the filled squares. Place the remaining 4 empty pastry squares directly over the filling, aligning the edges perfectly.

Press down around the outer edges with your fingers to seal. Use the floured tines of a table fork to crimp the edges tightly around the entire perimeter. Use a sharp paring knife to slice 2 small steam vents across the top of each hand pie.

Brush the tops of all 4 assembled pockets generously with the remaining egg wash. Bake for 20 to 25 minutes until the pastry layers rise high, displaying a glossy, deep golden-brown finish with bubbling cheese. Allow them to cool on the pan for 5 minutes before serving warm.