Preheat grill or broiler.


Cut each head of radicchio in half lengthwise and drizzle each half with ½ tablespoon olive oil and season well with salt and freshly ground black pepper. Place cut side down on the hot grill or face up under the broiler and cook without moving until just charred at the edges, about 4-5 minutes. Turn and cook for 3 minutes on the other side then remove and allow to cool.


Make the vinaigrette by whisking the vinegar and mustard together and then drizzling in the remaining oil to form a loose emulsion. Add the chives to the sauce and season the vinaigrette with salt. Using a potato peeler, shave 12 long pieces of the asiago and set aside.


To assemble place two halves cut side up on each plate, drizzle with 2 tablespoons of the vinaigrette and freshly grate horseradish overtop. Place 3 pieces of asiago over the radicchio on each plate and serve immediately at room temperature.