Heat oven to 375° F. Lightly oil a 9-by-13-inch or some other shallow 3-quart baking dish. In a large skillet, heat the oil over medium-high heat. Add the sausage and cook, breaking it up with a spoon, until browned, 6 to 8 minutes. Using a slotted spoon, transfer the sausage to a large bowl.
To the drippings in the skillet, add the celery, onions, apples, sage, ¾ teaspoon salt, and ½ teaspoon pepper. Cook, tossing often, until the vegetables are tender, 7 to 9 minutes.
Add the vegetable mixture, baguette, broth, and eggs to the sausage and toss to combine. Transfer to the prepared baking dish and cover with foil.
Bake the stuffing for 20 minutes. Remove the foil and bake until browned, 20 to 30 minutes more.