Preheat a grill or grill pan to medium-high heat.
Trim the base of the radicchio stems and cut each head into quarters through the core, keeping the core intact.
Brush each wedge generously with olive oil, making sure it reaches between the outer leaves.
Season the wedges with flaky salt immediately before grilling.
Place wedges cut-side down on the hot grill and cook undisturbed for 3–4 minutes.
Flip with tongs and grill the second side for another 3–4 minutes, until deeply charred and the core still has slight resistance.
While the radicchio grills, warm the balsamic vinegar in a small saucepan over medium heat and simmer for 4–5 minutes until reduced by half and syrupy.
Arrange the grilled wedges on a platter and let rest 3–4 minutes.
Drizzle with the reduced balsamic, then top with shaved parmesan and chopped toasted walnuts while still warm.