Preheat oven to 350 degrees F. In a large saute pan or dutch oven, heat the olive oil over medium heat. Brown the sausage until cooked through and reserve on the side.
Add the butter, apples, celery and onions and saute until tender, add the thyme leaves and return the sausage to the pan and stir thoroughly.
Transfer the mixture to a large bowl if there's not enough room in your pan to mix in the bread and the broth.
Thoroughly whisk the egg into the broth and combine the bread, broth mixture and sausage/vegetable mixture, season with salt and pepper. Put into a 9 x 13 baking pan. Cover with foil and bake about an hour, removing foil the last 10 minutes to ensure a brown and crunchy top.