Step 1Cut the leeks in quarters lengthwise almost to, but not through, the root end. Wash carefully under running water to remove all of the grit. Slice the quarters crosswise into eighth-inch slices.
Step 2In a large skillet, heat 3 tablespoons oil over medium heat in a large skillet. Add the leeks and the celery and cook until tender and the leeks begin to turn golden, about 5 minutes.
Step 3While the leeks are cooking, heat a medium skillet over medium-high heat. Add the remaining 1 tablespoon oil and the butter and heat until sizzling. Add the mushrooms and season with one-fourth teaspoon salt. Cook several minutes, tossing, until the mushrooms start to lose their water and darken. Add the minced shallot and cook, tossing, until the shallot softens, about 3 minutes. Remove from the heat and set aside.
Step 4Spoon the leek mixture into a large work bowl and add the walnuts, bread cubes, walnut oil and 2 cups water. Stir gently to combine without breaking up the cubes too much. If it seems at first there may be a little too much water, keep stirring until it is all absorbed.
Step 5Fold in the mushrooms and 1 1/2 teaspoons salt. Taste and add more salt if necessary.
Step 6Add the beaten eggs, stirring until they are no longer visible among the bread cubes.
Step 7Transfer the mixture to a glass or earthenware covered baking dish. Cut an aluminum foil square just big enough to fit inside the opening of the dish and smear it with butter. Press it down lightly on top of the stuffing, butter-side down. Cover the baking dish with the lid and place it in a cold oven. Turn the oven to 350 degrees and bake until the stuffing is hot and steaming, about 30 minutes. Remove the lid and foil, checking to see if the top is golden; if not, return the stuffing to the oven, uncovered, and bake an additional 10 to 15 minutes until crisp and golden on top.
Step 8Remove the dish from the oven and keep covered until you're ready to serve.