2 1/2 to 3 pounds winter squash
1/4 cup olive oil, plus extra for the squash
6 cloves garlic, unpeeled
12 whole sage leaves, plus 2 tablespoons chopped
2 onions, finely chopped
Chopped leaves from 4 thyme sprigs or 1/4 teaspoon dried thyme
1/4 cup chopped parsley salt
freshly milled pepper
2 quarts water or stock
1/2 cup Fontina, pecorino, or ricotta salata, diced into small cubes
Preheat the oven to 375°. Halve the squash and scoop out the seeds. Brush the surfaces with oil, stuff the cavities with the garlic, and place them cut sides down on a baking sheet. Bake until tender when pressed with a finger, about 30 minutes.
Meanwhile, in a small skillet, heat the 1/4 cup oil until nearly smoking, then drop in the whole sage leaves and fry until speckled and dark, about 1 minute. Set the leaves aside on a paper towel and transfer the oil to a wide soup pot. Add the onions, chopped sage, thyme, and parsley and cook over medium heat until the onions have begun to brown around the edges, 12 to 15 minutes.
Scoop the squash flesh into the pot along with any juices that have accumulated in the pan. Peel the garlic and add it to the pot along with 1 1/2 teaspoons salt and the water and bring to a boil. Lower the heat and simmer, partially covered, for 25 minutes. If the soup becomes too thick, simply add more water to thin it out. Taste for salt.
Depending on the type of squash you've used, the soup will be smooth or rough. Purée or pass it through a food mill if you want a more refined soup. Ladle it into bowls and distribute the cheese over the top. Garnish each bowl with the fried sage leaves, add pepper, and serve.