Divide the pierogi dough in half. Wrap the piece you're not working with and set aside.
Roll the dough about ⅛" thick. You don't have to be super precise with this, but it should be pretty thin but not feel like it's going to tear on you. If it becomes see-through, you've rolled too far.Dust additional flour down as needed to prevent sticking, and rotate or flip the dough frequently as you roll.If the dough fights you or shrinks back, cover it with a towel and let it rest for 3-5 minutes, then resume rolling. Use the bottom of the 5" pierogi press to cut as many circles as you can out of the dough. Knead the dough scraps back into a ball, wrap in plastic, and set aside for later.
Working one pierogi wrapper at a time, roll each dough circle a little bit bigger to make room for the filling. Place the rolling pin in the center of the circle and roll outward, rotating 90 degrees each time to keep the circle even.
Align the circle of dough in the pierogi press. There will be some hanging over, but try to get one half of it as perfectly aligned as possible. That will be the "bottom" side of the pierogi.
Scoop 3 tablespoon of the stuffing mixture into your hands and shape it into a half-moon/log. Press it into the bottom half of the pierogi press and gently smush so it fits.
Fold the other half of the dough round up over the filling, align it with the bottom layer of dough, and gently press down with your fingers so it stays in place.If the dough seems dry, brush the bottom edge with a few drops of water. Slowly fold the empty half of the press over the pierogi. Adjust the pierogi wrapper as needed if it shifts, particularly in the corners by the hinges. Press it firmly shut for a few seconds, then release. Gently remove the pierogi and set it aside on a parchment lined sheet pan or sheet pan dusted lightly with corn starch. Repeat with remaining pierogi dough and filling. Cook immediately or freeze until ready to use.