1. Pour oil into a 6- to 8-inch frying pan over medium-high heat. When hot, add whole sage leaves and stir until they turn darker green, 45 seconds to 1 minute. With a slotted spoon, lift out leaves and drain on paper towels. Reserve oil.
2. Rinse squash and, if whole, cut in half and scoop out and discard seeds. Brush cut surfaces of squash and onions with a little of the reserved oil. Lay vegetables, cut side down, in a 10- by 15-inch pan. Slip garlic under squash.
3. Bake in a 375° oven until all vegetables are soft when pressed, 45 to 60 minutes. Reserve juices. Scoop squash from skin. Peel and chop onions. Peel garlic. Discard skin and peels.
4. In a 3- to 4-quart pan, combine squash, onions, garlic, parsley, chopped sage, and thyme. Mash squash mixture with a potato masher. Stir in reserved juices and broth. Stirring often, bring to a boil over high heat. Cover, reduce heat to low, and stir occasionally to blend flavors, 15 to 20 minutes.
5. Add the remaining sage cooking oil and season soup with salt to taste. Before serving, distribute cheese equally among individual bowls. Add soup, top with fried sage, and sprinkle with pepper.