Cut the radicchio in quarters lengthwise, being sure to keep some of the stem attached to each quarter. Trim off any dark bits of stem. Submerge the radicchio quarters in ice water for 1 hour to remove some bitterness. Put a plate on top of the radicchio to keep them under water.
In a medium bowl, combine all ingredients except the mozzarella. Mix well.
Prepare a hot fire in your grill, or heat a cast iron grill pan over medium-high heat.
Drain the radicchio and place them on paper towels to absorb remaining water. Open up the leaves and spoon the olive oil mixture inside. Place the radicchio quarters on a baking sheet and pour the remaining marinade mixture. Season generously with salt and pepper.
Grill the radicchio until browned on the outside but still raw in the center, 3 to 5 minutes, turning occasionally. Be careful, as the oil mixture may flame up.
When the radicchio begins to brown, pull from the grill, open up a section of each portion and place a slice of mozzarella inside. Place quarters back onto the sheet pan and place the pan directly on the grill for 5 minutes to melt the cheese. Drizzle any leftover dressing over the top of the grilled radicchio and serve warm.
If making radicchio in advance: Before the party, have the radicchio drained and marinated in the vinaigrette in the refrigerator.