Heat the oven to 350 degrees F. Lightly butter an 8 by 10-inch gratin dish.
Put the milk, onion, thyme and cloves in a small saucepan and bring almost to a boil, then remove from the heat to steep while you prepare the radicchio.
Cut the radicchio into small squares or narrow ribbons. Heat the oil in a wide skillet over medium-high heat. Add the radicchio and cook, stirring occasionally, until it has wilted and browned, about 4 minutes. Season with salt and pepper and remove from the heat.
Strain the milk and discard the aromatics. Melt the butter in a small saucepan over medium heat. When the butter foams, add the flour. Cook while stirring for about 1 minute, then whisk in the warm milk. Stir constantly, until thickened, about 2 minutes, then add 3/4 cup of the cheese, and stir until well mixed. Transfer the mixture to the waiting gratin dish and smooth the top. Cover with the bread crumbs and the remaining cheese.
Bake until the bread crumbs have crisped and browned and the gratin is bubbling, about 25 minutes. Let the gratin settle for about 10 minutes before serving.