Coarsely purée all the ingredients for the salsa verde, except the oil, in a food processor or blender. With the motor still running, gradually trickle in the oil to make a creamy paste. Transfer to a serving dish, cover and set aside. (The salsa will keep for up to 1 week in the refrigerator.)
Heat the grill on the hottest setting. Arrange the chicory halves on the grill rack, cut-sides down, brush with some of the oil and cook under the grill for 5 minutes. Turn the chicory halves over, brush with the remaining oil and sprinkle the Parmesan over the top. Cook for a further 4 minutes or until the cheese has melted and the edges of the chicory begin to char.
Transfer the chicory to plates and garnish with chopped parsley. Add a little salsa verde to each plate and serve immediately, offering the remaining salsa verde separately. Toasted ciabatta bread is a good accompaniment.