Slice the radicchio in half lengthways. Set 1 piece aside.
Slice the other piece crossways into strips.
Heat the oil and half the butter in a wide shallow pan.
Cook the onions and garlic until soft, then add the radicchio strips and the rice and cook, stirring until the rice is translucent.
Pour in the wine and cook until absorbed.
Add the stock, a ladleful at a time and cook until absorbed before adding more.
Continue, stirring, until all the stock is used.
Stir constantly until the rice is almost cooked, but still with a slight bite.
Heat the broiler (grill).
Add the thyme, cheese and remaining butter to the risotto and season to taste with salt and pepper.
Season the reserved piece of radicchio with salt and pepper and drizzle with olive oil. Place on the broiler and cook, turning, for 7-10 minutes, until tender, reducing the heat towards the end of cooking time.
Slice the grilled radicchio and serve on top of the rice. Serve with Parmesan.