Heat the oven to 350°F and butter a 9-by-13-inch baking dish with 1 teaspoon of the butter.
Brown the sausage in a large skillet over medium-high heat for 7 to 9 minutes, breaking it into crumbles, until it reaches 160°F, then transfer it to a large bowl.
Melt the remaining butter in the skillet and cook the onion, celery and apples for 8 to 10 minutes, until the onion is soft and the apples hold their shape.
Add the vegetables, bread cubes, sage, thyme, salt and pepper to the sausage and toss well.
Whisk the broth and eggs together, pour them over the bread mixture and toss until evenly dampened.
Spread the mixture in the dish, cover with foil and bake for 30 minutes.
Uncover and bake for 15 to 20 minutes more, until crisp on top and 160°F in the center, then rest for 10 minutes.