Cook Mushroom Filling: Heat 1 tablespoon of vegetable oil in a large skillet over medium heat. Add the chopped onion and mushrooms, then cook for 8 to 10 minutes, stirring often, until the mushrooms are soft and the mixture looks golden.
Season Filling: Add salt and black pepper to taste. Stir well, then remove the mushroom mixture from the heat and let it cool slightly.
Make Potato Dough: Place the cold mashed potatoes in a mixing bowl. Add the egg, all-purpose flour, and ¼ cup cornmeal, then mix until the potato mixture holds together like a soft dough.
Season Potato Dough: Taste the potato mixture and add a little salt and black pepper if needed. Mix again until evenly combined.
Shape Potato Disk: Sprinkle some of the remaining cornmeal onto a clean work surface or plate. Scoop about ⅓ cup of the potato mixture and flatten it into a round disk.
Add Mushroom Filling: Place about 1 tablespoon of the mushroom filling in the center of the potato disk. Do not overfill, or the pancake may split while cooking.
Seal Pancake: Fold the potato mixture around the filling and gently pinch the edges closed. Shape it into a thick, flat pancake.
Coat Pancake: Coat both sides of the stuffed pancake with cornmeal. Repeat with the remaining potato mixture and mushroom filling.
Fry Pancakes: Heat the remaining 2 tablespoons of vegetable oil in the skillet over medium heat. Cook the pancakes for 3 to 4 minutes per side, or until golden brown and crisp.
Drain Pancakes: Transfer the cooked pancakes to a plate lined with paper towels. Let them rest for 1 to 2 minutes so they firm up slightly.
Serve: Serve the potato pancakes warm with sour cream and chopped green onions.