1. In a heatproof bowl, soak the morels in the boiling water until softened, about 20 minutes. Rub them to loosen any grit, then lift them out and coarsely chop.
2. In a large bowl, combine the flour with 1/2 teaspoon of salt. Add the eggs and stir until the dough just comes together. Transfer to a lightly floured work surface and knead briefly until soft and smooth. Wrap the dough in plastic and set aside at room temperature for 30 minutes.
3. Meanwhile, cook the potatoes in a large saucepan of boiling salted water until tender, about 20 minutes; drain. Return the potatoes to the pan and shake over high heat to dry them, about 1 minute. Using a potato masher, mash the potatoes. Mash in the milk, sour cream, Parmesan and butter. Stir in the morels and season with salt and pepper. Transfer to a bowl and let cool slightly.
4. Cut the dough in half and pat each piece into a disk. On a well-floured work surface, roll out the disks to 9-inch rounds. Cover with plastic wrap and let stand for 15 minutes.
5. On a floured work surface, roll out 1 round at a time to a 1/16-inch thickness. Using a 4-inch biscuit cutter, stamp out 9 smaller rounds. Lightly moisten the edge of 1 round and dollop 1 rounded tablespoon of the potato filling on half of it. Pat the filling slightly; fold the dough over the filling to form a half moon and press the edges to seal. Set aside on a floured platter. Repeat with the rest of the dough and filling.
6. Cook half the pierogies at a time in a large pot of boiling salted water until they rise to the surface, about 1 minute. Using a slotted spoon, transfer them to paper towels and dry. In a large bowl, toss the pierogies with just enough vegetable oil to coat.
7. Light a grill or heat a grill pan. Grill the pierogies over a hot fire, turning once, until browned, about 3 minutes per side. Remove from the grill. Add the celery leaves and grill until lightly charred, about 1 minute per side. Transfer the pierogies to a platter and garnish with the grilled celery leaves.
Make Ahead: The pierogies can be prepared through Step 5 and frozen for up to 1 month. Boil the frozen pierogies for the same amount of time indicated for the fresh ones.
Wine Recommendation: The rich, smoky filling in these pierogies makes them perfect foils for a full-flavored Merlot with velvety tannins like the 1994 Columbia Crest Estate Series from Washington State or the 1995 Meerlust from South Africa.