Heat the oven to 200 C / 400 F. Line a sheet pan (33 x 46 cm / 13 x 18 inch) with parchment paper.
Trim the base of each radicchio head just enough to keep the core intact, then cut each head into 4 wedges through the core so the leaves stay attached.
In a mixing bowl, whisk together the olive oil, 3 tbsp of the balsamic vinegar, honey, salt, and pepper.
Arrange the radicchio wedges cut-side up on the sheet pan, spaced apart so they roast rather than steam. Brush both cut sides with the balsamic mixture using a pastry brush. Scatter the garlic slices and thyme sprigs over the top.
Bake for 12 minutes, then flip each wedge carefully with tongs and brush with any remaining marinade.
Return to the oven and bake another 10 to 13 minutes, until the outer leaves are charred at the edges and a knife tip slides into the core with slight resistance.
Remove from the oven and discard the thyme sprigs. Drizzle the remaining 1 tbsp balsamic vinegar over the hot wedges.
Shave the parmesan over the top with a vegetable peeler while the radicchio is still warm. Serve right away.