Preheat an outdoor grill for medium heat and lightly oil the grate.
Pour 1/2 tablespoon olive oil onto a plate. Place radicchio cut-side down onto the plate to coat in oil. Pour remaining 1 1/2 tablespoons olive oil in a bowl; add mushrooms, walnuts, and cashews and toss to coat.
Place oiled radicchio cut-side down onto the preheated grill and cook to desired doneness, 3 to 5 minutes.
Meanwhile, pour mushroom-nut mix into a grilling pan. Cook on the preheated grill until mushrooms have softened, 3 to 5 minutes.
Let grilled radicchio halves cool slightly; cut in half again to make quarters and place on a serving plate. Top evenly with mushroom mixture and blue cheese dressing.