Set a wire rack over a baking sheet. Bring a large pot of water to a boil. Add salt, return to a boil, and add greens; cook 1 minute. Transfer greens to wire rack to cool. Transfer baking sheet to refrigerator and let greens chill for 20 minutes.
Gently squeeze greens to remove liquid. Finely chop greens, place in a kitchen towel, and wring out any remaining liquid.
In a medium bowl, mix together Parmigiano-Reggiano and ricotta cheese. Add greens; season with salt and pepper. Mash filling together using a wooden spoon. Filling should be moist enough to hold together when pressed. Filling can be kept refrigerated, in a covered container, up to 1 day.
Divide pierogi dough evenly into 4 pieces; wrap in plastic wrap. Sprinkle work surface with flour. Working with 1 piece of dough at a time, unwrap dough and roll out to about 1/4 inch thick. Feed dough through the largest setting on a pasta machine. Continue rolling dough through machine, decreasing the opening size each time until dough is about 1/16 inch thick. Place dough on work surface and, using a 4-inch round cookie cutter, cut circles from dough. Cover circles with plastic wrap or a kitchen towel while repeating process with remaining dough. You should have 12 to 14 circles total.
Lightly flour a baking sheet; set aside. Place a scant tablespoon of filling just to one side of the center of each circle of dough. Working with one pierogi at a time, brush the edges of the circle with a damp pastry brush. Fold circle in half to enclose filling, pressing sides together firmly so no air is trapped. Seal edges by pressing down with tines of a fork. Transfer pierogi to prepared baking sheet and keep covered until ready to cook. Repeat process with remaining dough and filling.
Bring a large pot of salted water to a boil. Carefully add the pierogi, stirring gently so they do not stick to the bottom of the pot. Return to a boil and cook until the dough edge no longer feels stiff when pinched and looks slightly translucent, 7 to 8 minutes. Using a slotted spoon, transfer pierogi to a kitchen towel to drain.
Place pierogi, brown butter sauce, and 1 tablespoon parsley in a large skillet; toss to combine. Place skillet over high heat and cook until heated through, about 1 minute.
Divide beets evenly between 4 plates. Place 3 pierogi on top of beets; spoon over sauce. Garnish with ricotta salata and sprinkle with remaining teaspoon parsley.