Preheat oven to 375 degrees. Brush a 2 1/2-quart baking dish with butter.
Melt 2 tablespoons butter in a large saute pan over medium heat. Add onion, leek, and 1/2 teaspoon salt, and cook, stirring occasionally, until softened, 7 to 8 minutes. Stir in thyme, and transfer to a plate.
Return pan to stove, raise heat to medium-high, and melt 2 tablespoons butter. Saute half the mushrooms with 1/2 teaspoon salt, stirring occasionally, until golden brown, 3 to 4 minutes. Transfer to plate with onion and leeks. Saute remaining mushrooms in same manner, and transfer to plate.
Return pan to stove over medium-high heat. Add wine, bring to a boil, and cook until reduced by half. Add broth, and bring to a simmer.
Place bread in a large bowl. Pour broth mixture over bread, and toss until evenly absorbed. Gently stir in mushrooms, onion and leeks, nuts, and eggs; season with pepper. Stuffing can be made up to this point (do not add eggs) 1 day ahead and refrigerated. Let come to room temperature before stirring in eggs and baking.
Transfer dressing to baking dish. Cover, and bake 30 minutes. Uncover dish, dot with remaining 2 tablespoons butter, and bake until top is golden brown and crisp, 30 to 35 minutes. Let cool 30 minutes before serving.