Preheat the oven and cook sausage. Preheat oven to 375°F. Heat oil in a large skillet over medium-high heat. Cook the sausage, breaking it up with a spoon until browned, 6 to 8 minutes. Transfer to a plate, reserving drippings in pan.
Toast bread in the oven. Divide the bread between 2 large rimmed baking sheets. Bake until dry and crisp, 10 to 12 minutes.
Cook vegetables on the stove-top. Meanwhile, melt butter in reserved skillet over medium heat. Add onions, celery, pepper, and 1 teaspoon salt. Cook, stirring occasionally and scraping any browned bits from bottom of skillet, until very tender and starting to brown, 10 to 12 minutes. Add wine and cook, stirring occasionally, until evaporated, 2 to 4 minutes. Transfer to a large bowl and let cool for 10 minutes.
Assemble the stuffing, cover, and bake. Coat a 13-by-9-inch baking dish with butter. Add toasted bread, sausage, broth, parsley, sage, eggs, and remaining ½ teaspoon salt to vegetables in bowl; toss gently to combine. Transfer to prepared baking dish. Cover with aluminum foil and bake for 20 minutes.
Uncover dish and finish baking. Uncover dish and bake until browned, 20 to 30 minutes more.
