Prepare the filling ingredients by mincing the garlic, slicing the green onion, grating the Parmesan, and draining and squeezing the vegetables, with softened cream cheese and room-temperature eggs ready nearby. The vegetables should feel damp rather than wet, and the cream cheese should yield easily when pressed.
Mash the cream cheese with a fork in the medium bowl, working in one egg, the garlic, oregano, and black pepper. A cohesive mixture without firm cream cheese lumps will spread across the pastry without tugging or tearing it.
Fold the green onion, feta, spinach, artichokes, and Parmesan into the cream cheese mixture, then set the bowl aside. Keep stirring until the vegetables and cheese appear evenly distributed, with no dense pockets of spinach.
Open one pastry sheet on parchment, rolling only as needed to form a square approximately 10 x 10 or 12 x 12 inches, depending on the sheet’s size. Pastry already supplied in a suitable sheet needs no additional rolling and should remain cool, flexible, and evenly thick.
Cover the pastry with a thin, even coating of filling, reserving enough mixture for the second sheet. A level layer creates a more consistent spiral than thick patches that push outward when you roll.
Lift the near edge of the parchment to begin curling the pastry, then guide it into a log with moderate tension and repeat with the remaining pastry and filling. The coil should hold its shape without compressing the dough, leaving the inner pastry layers room to expand.
Beat the remaining egg, apply a light coating around each log, and cut crosswise into rounds about 1/2 inch thick using a sharp or serrated knife. Gentle cutting strokes preserve the spiral better than pressing straight down through soft dough.
Lay the rounds flat on parchment-lined baking sheets with about 2 inches between them, then bake in a preheated 400°F oven for 18 to 20 minutes. Look for expanded pastry layers, golden-brown edges, and a crisp-looking surface rather than relying on the timer alone.
Leave the baked pinwheels on their sheets for 5 minutes before lifting them with a thin spatula. The filling will settle slightly, and any cheese that escaped onto the parchment may form delicate, lacy frico edges that lift more easily after cooling.