Preheat oven to 350° F with the rack in the middle or upper middle. Generously butter a 9 by 13 inch baking dish.

Chop the onion, celery, and carrot fine in a food processor. Sauté in butter until very soft but not browned, about 10 minutes.

Add poultry seasoning, parsley, salt, and pepper. Stir to combine. Turn off heat and let cool slightly before mixing with the eggs.

Whisk together eggs, milk, and chicken broth in a medium bowl.

In a large bowl or large baking pan, combine bread cubes, sautéed vegetables, and milk mixture. Gently toss together with your hands or a spoon until combined. Don’t overmix and take care not to break up the bread cubes.

Let the mixture stand for a few minutes until the bread has absorbed the liquid, then pour into prepared baking dish. Cover with foil and bake for 30 minutes.

Remove the foil, brush the top of the dressing with melted butter, and return to oven.

Bake until the top is a toasty, golden brown and getting crisp. For an extra crispy top, put under the broiler for a minute or two.

Remove from oven and serve warm.