Cut a slit in each chestnut with scissors or a paring knife. Cook chestnuts in a pot of boiling water for 20 minutes, then drain in a colander. When cool enough to handle, peel off and discard shells and inner brown skins.
Heat 3 tablespoons olive oil in a large saute pan over medium high until shimmering. Saute chestnuts, stirring occasionally, until they begin to brown. With a slotted spoon, transfer chestnuts to a plate. Add both mushrooms to skillet and saute, stirring occasionally, until mushrooms release their juices and begin to turn golden, about 15 minutes. Transfer to a large bowl and season with salt. Add bread cubes, and toss to mix.
Heat 3 tablespoons olive oil in same pan over medium high. Add carrots, onions, celery, sweet potato, and garlic; season with salt and pepper. Cook, stirring occasionally, until vegetables begin to brown, about 10 minutes. Add stock and simmer until vegetables are tender, about 5 minutes. Transfer contents of pan to bread mixture along with the chestnuts. Add apples, herbs, and another 3 tablespoons olive oil. Toss well and season stuffing with salt and pepper, then mix in the eggs.
Preheat oven to 375 degrees. Remove top of pumpkin by cutting a circle about 5 inches in diameter around stem with a paring knife: reserve top. Scrape out and discard seeds and stringy flesh from pumpkin. Rub inside of pumpkin with remaining 1 tablespoon olive oil and season with salt and pepper. Spoon stuffing into pumpkin and replace top. Bake stuffed pumpkin on a rimmed baking sheet until pumpkin is tender when pierced with a skewer, 1 1/2 to 2 hours. (Do not overcook pumpkin or the side may split.) To serve, remove top and scoop out stuffing and some pumpkin flesh with a large spoon.
A cheese pumpkin is the same color as butternut squash, and it tends to be flatter than the more common sugar pumpkin. The flesh is quite thick and very smooth in texture. If this kind of pumpkin is not available, use what you can find, but check for doneness after 1 hour 15 minutes, since the flesh will most likely not be as thick, and the pumpkin will cook more quickly.