Slice the radicchio in half lengthwise and add to a mixing bowl with the balsamic vinegar and olive oil. Use your hands to coat the vegetables in the oil and vinegar, then set aside for 15 minutes to marinade.
Preheat the oven to 395 degrees Fahrenheit.
Remove the radicchio from the marinade, leaving the excess liquid to one side. Place the radicchio cut side down on a shallow oven tray and roast for 7 minutes until tender.
Turn the radicchio over so that it is cut side up and season with a little salt.
Dividing evenly between the four halves, crumble the goats cheese over the top of the radicchio.
Roughly chop the walnuts and scatter them over the radicchio.
Return to the oven to bake for a further 3-5 minutes until charred and tender and the cheese is melted.
Pour the reserved oil and vinegar over the radicchio and serve immediately.