Preheat the oven to 350°F.
Rub the inside of the kabocha squash with olive oil and sprinkle with salt and pepper. Place flesh-side down on a sheet tray lined with aluminum foil and bake until cooked through (check with the tip of a paring knife), 40 minutes to 1 hour. Set aside to cool.
Turn the cheese pumpkin upside down and, using a sharp knife, carefully remove a circular cap approximately 10 inches in diameter. Cut the sourdough bread into 1-inch-thick slices and toast. Lightly rub the toasted bread with the cut side of the garlic, then dice each slice and place the pieces in a large mixing bowl.
In a medium saute pan over medium heat, add the bacon and cook, stirring, until crispy. Add the onions and cook, stirring occasionally, until tender. Toss in the mushrooms and cook, stirring occasionally, for about 3 minutes, then season with salt and pepper. Transfer the mixture to the bowl with the diced bread.
When the cooked kabocha squash is cool enough to handle, remove the flesh with a spoon and cut into a rough dice. Add the squash, Gruyere, Fontina and walnuts to the bowl with the bread and mix the stuffing well.
In a separate bowl, whisk together the half-and-half with the ground spices, sage and salt and pepper to taste. Sprinkle the inside of the pumpkin with salt and pepper and fill the cavity with the stuffing.
Pour the half-and-half mixture into the pumpkin until the liquid reaches the rim. Return the cap to the pumpkin and place it on a foil-lined baking tray. Bake until the pumpkin is cooked through, 1 ½ to 2 hours. Serve warm, scooping out a bit of the pumpkin flesh from the sides, along with the stuffing.
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