To prepare stuffed pumpkin: Place rack in center of oven; preheat to 350 degrees F. Line a rimmed baking sheet with parchment paper.

Using a sharp knife and working at a slight angle, cut a 5- to 6-inch diameter cap off the top of the pumpkin (or squash)--just like a jack-o'-lantern. (If necessary, cut a small slice off the bottom so it rests flat.) Using a metal spoon, remove the seeds and strings from the cap and the inside. (If preparing Spiced Pumpkin Seed garnish, reserve the seeds.) Season the inside with 1/2 teaspoon each salt and pepper. Place the pumpkin (or squash) on the prepared baking sheet.

Place bread in a bowl. Heat 1 tablespoon oil in a large skillet over medium heat. Add onion and celery. Cook, stirring, until just beginning to brown, 3 to 5 minutes. Reduce the heat to medium-low, add mushrooms and season with 1/4 teaspoon each salt and pepper. Cook, stirring occasionally, until the mushrooms are tender, 6 to 8 minutes. Add garlic and cook until fragrant, about 45 seconds. Transfer the vegetables to the bowl with the bread. Stir in thyme, sage, cayenne (if using) and the remaining 1/4 teaspoon pepper. Add eggs, milk and cheese and stir to combine. Fill the pumpkin (or squash) with the mixture, pushing it down if necessary so it fits inside. (If you have extra stuffing, place it in a small baking dish, cover with foil and bake during the last hour that the pumpkin roasts.) Replace the cap on top.

Bake for 1 1/2 hours.

Meanwhile, if preparing Spiced Pumpkin Seed Garnish: Line a rimmed baking sheet with a nonstick baking mat or parchment paper. Rinse the reserved pumpkins seeds to remove any flesh or strings. Dry them on a clean dish towel. Toss in a small bowl with 1 teaspoon oil, fennel seed, crushed red pepper and 1/8 teaspoon each salt and pepper. Spread out on the prepared baking sheet. Roast at 350 degrees F until golden brown, about 20 minutes.

After the pumpkin has baked for 1 1/2 hours, remove the cap and use a spoon to fluff up the stuffing so it comes up higher than the top of the pumpkin (or squash). Return to the oven (without the cap) and continue baking until it is tender enough to be pierced easily with the tip of a knife (check in several spots to be sure it's done) and an instant-read thermometer inserted in the center of the stuffing registers at least 160 degrees F, 30 to 45 minutes more. Let rest for 10 minutes. Carefully transfer to a warmed serving plate. Garnish with Spiced Pumpkin Seeds, if desired.