Prepare the filling. Place the diced shallots and clarified butter in a large pan over low heat. Cook, stirring frequently, until the shallots are translucent, about 4 minutes. Add the sour cabbage, mashed potatoes, and mushrooms. Season with salt and pepper. Fry for 10 minutes, stirring occasionally. Remove from heat. Drain any remaining liquid from the filling, using a colander, and place the filling in a food processor. Pulse until finely chopped and smooth. Set aside.
Bring a large pot of salted water to a boil.
Sift the flours into a large bowl and add sea salt and 4 tablespoons butter. Mix with a spoon. Add hot water and mix well to combine. The dough will be relatively smooth but still a bit sticky. Turn it onto a lightly floured surface and knead for 2-3 minutes, until firm.
Roll out the dough on a lightly floured surface until ⅛-inch thick. Work fast to prevent the dough from getting dry.
Using a glass, cut out circles from the dough. Place 1 teaspoon of the filling in the center of each circle. Holding 1 circle in your hand, fold the dough over the filling. Pinch the edges and repeat with the remaining ingredients. Working in batches, transfer pierogies to boiling water. Cook through, about 4-5 minutes. Remove from the pot using a slotted spoon; drain and set aside. Continue until all the pierogies are cooked.
Fry the drained pierogies in butter in a large skillet until browned, about 2-3 minutes per side. Decorate with fresh oregano leaves and serve immediately.