Horizontally insert bamboo skewer into radicchio wedge to hold it together. Brush radicchio with oil. Sprinkle with 2 tablespoons plus 2 teaspoons of the vinegar. Set aside.
Brush remaining vinegar on both sides of MORNINGSTAR FARMS (R) GRILLERS (R) ORIGINAL veggie burger. Grill burger according to package directions. Add radicchio to grill during the last 6 minutes of cooking time. Grill just until radicchio wilts and begins to brown, turning frequently.
Coarsely chop radicchio. Serve burgers, topped with radicchio, cheese and basil leaves in buns.