Set the pumpkin in hot coals at the base of a campfire or in a portable firepit located on a nonflammable surface at least 15 feet away from any structures or woody material. Pile up the coals around the pumpkin. Roast until tender, about 20–30 minutes. Don’t worry about the exterior char; move the pumpkin around while roasting so it cooks evenly.
Carefully remove the pumpkin from the coals, let it cool to the touch (10–15 minutes), then remove the top with a knife, jack-o’-lantern style, and scoop out the seeds.
Place the pumpkin in a small-to-medium (~2 quart) cast-iron dutch oven and return to the coals. Place the lid on and keep warm while you prepare the mushrooms.
Set a cast-iron skillet over low flames on a grill grate or directly on the coals. Add the butter and shiitakes. Sauté until deeply caramelized, add the garlic, and cook for a minute longer. Add a pinch of salt.
Remove the lid from the dutch oven. Add wine into the pumpkin, then slowly stir in cheese a handful at a time, waiting for each batch to melt before adding the next.
Once all the cheese has been added, top with the mushrooms and remove from the fire. Eat with spoons, scooping out chunks of soft, roasted pumpkin and melted cheese. Serve with a crusty French baguette.