Make the venison: Heat the oven to 350°. In a large enameled cast-iron pot, heat the oil over medium-high. Season the venison all over with salt and pepper and add to the pot and cook, turning as needed, until golden brown on all sides, about 18 minutes. Transfer the venison to a plate and add the celery, onions, and garlic to the pot. Cook, stirring, until the vegetables are soft and beginning to caramelize, 8 to 10 minutes. Pour the wine into the pot and cook, stirring, until reduced by half, about 6 minutes. Return the venison to the pot along with the stock and carrots and bring to a boil. Tie the parsley and thyme together with kitchen twine and add to the pot along with the mustard. Cover and bake until the venison is very tender, about 4 hours.

Meanwhile, make the pierogi dough: In a large bowl, whisk the flour with the ½ teaspoon salt and create a well in the center of the flour. Add 2 eggs and ⅓ cup water to the well and stir, slowly adding flour, until the dough comes together. Transfer the dough to a work surface and knead briefly until smooth. Place the dough ball on a lightly floured baking sheet and cover with plastic wrap. Let the dough stand at room temperature for 30 minutes.

In a 12-inch skillet, heat the olive oil over medium-high. Add the mushrooms and shallots, season with salt, and cook, stirring, until all the mushrooms' moisture evaporates and they begin to brown, about 10 minutes. Remove the skillet from the heat and let the mushrooms cool to room temperature. Scrape the mushrooms into a bowl, stir in the cheese and lemon juice, and season the filling with salt and pepper.

In a small bowl, whisk the remaining egg with the milk to make an egg wash. Transfer the pierogi dough to a lightly floured work surface and roll until ⅛ inch thick. Using a 3 ¾-inch round cutter, cut out circles of dough. Place 1 tablespoon of filling in the center of each circle, moisten the edge of the circle with the egg wash, fold the circle in half, and seal at the edges.

Bring a large pot of salted water to a boil and, working in batches, add the pierogi and cook until the dough is no longer raw, about 2 minutes. Using a slotted spoon, transfer the pierogi to a sheet of parchment paper and let dry for 5 minutes. In a 12-inch skillet, heat the butter over medium-high. Working in batches, add the pierogi and cook, turning once, until golden brown on both sides, 2 to 3 minutes.

To serve, remove the venison from the pot and cut into bite-size pieces. Divide the venison among serving bowls and ladle the vegetables and cooking liquid on top. Place the pierogi alongside the venison in the bowls and sprinkle with thyme leaves.