Preheat oven to 375°F with a rack in the center.
Melt the butter over medium-high heat in a large frying pan.
Add onion and celery and sprinkle with ¼ teaspoon of the salt. Cook, stirring occasionally, until beginning to soften and turn translucent, about 5 minutes. Adjust heat as necessary to prevent browning.
Add apple, chestnuts, thyme, and sage, and continue cooking until apple has softened, about 5 minutes more.
Place bread cubes into a large mixing bowl and scrape the contents of the frying pan overtop.
Pour in the broth and the beaten eggs. Sprinkle with remaining ¼ teaspoon salt and the pepper.
Mix gently until everything is well-distributed, then let the mixture sit for a few minutes so the bread cubes absorb the wet ingredients.
Transfer stuffing to a large baking dish such as an extra-deep 9x13-inch pan or a 5-quart braiser.
Bake uncovered for 45 minutes, until crisp on top and tender and custardy inside.