Preheat the oven to 400 (F).
To prepare squash: Peel the squash, then halve and scoop out seeds. Cut into 1/2-inch to 3/4-inch chunks. Pour 2 tablespoons of olive oil into one or two smaller baking sheets, spreading it to an even slick. Lay the squash chunks on the baking sheet in one layer, sprinkle with 1/2 teaspoon of the salt, and a couple turns of freshly ground black pepper, and roast in the preheated oven for 30 minutes, or until squash is tender, turning the pieces occasionally so that they brown evenly. Set aside to cool slightly. Leave the oven on.
While the squash is roasting, melt 1 tablespoon butter and remaining 1 tablespoon of olive oil in a heavy frying pan, and cook the onions over medium-low heat with the remaining teaspoon of salt, stirring occasionally, until soft and tender, about 25 minutes. Stir in the cayenne pepper.
When the squash and onions are finished, mix them together in a medium bowl along with 2 cups grated fontina cheese and 1 teaspoon chopped fresh thyme.