Scott Conant says that fonduta (melted cheese sauce) is traditional throughout the Piedmont region and also popular in Valle d' Aosta, to the north. Fonduta is laced with white truffles when they're in season, making it superbly rich and luxurious.
Light a grill. Brush the zucchini, yellow squash, portobellos, tomatoes and onion with the oil and season with salt and pepper. Sprinkle with the thyme and grill over high heat, turning occasionally, until the vegetables are lightly charred, about 8 minutes. Transfer to a platter and keep warm.
Meanwhile, in a saucepan, bring the milk and cream to a boil. Add the fontina and cook over low heat, stirring, until creamy, 2 minutes. Stir in the Parmigiano-Reggiano. Strain the fonduta through a fine sieve and pour it into a pitcher. Season with salt and pepper. Sprinkle the vegetables with the chives and serve the fonduta on the side.