1. In a large skillet, heat 1 tablespoon oil over medium-high. Add onions and celery; season with salt and pepper. Cook, stirring occasionally, until vegetables are soft, 5 to 7 minutes; transfer to a large bowl. Add 1 tablespoon oil, mushrooms, and sage to skillet; season with salt and pepper and cook, stirring often, until mushrooms are browned, 5 minutes. Add wine and cook, stirring and scraping up any browned bits with a wooden spoon, until wine is almost evaporated. Transfer to bowl with vegetables; add walnuts, bread, and enough broth to moisten (stuffing should feel moist but not soggy). Season with salt and pepper and toss to combine.

2. Add eggs to bread mixture and toss well to combine. Reserve 4 cups stuffing for turkey. Spoon remaining stuffing into a lightly oiled 2-quart baking dish. (To store, cover and refrigerate, up to 1 day.) Preheat oven to 400 degrees, with rack in upper third. Bake, uncovered, until golden brown on top, 30 minutes.