Cut each head of radicchio in half lengthwise and place them, cut side up, in an 11 x 16-inch baking dish. Remove the Portobello stems and cut them in half lengthwise. Cut the caps in half, and place the caps and stems in the same baking dish. Be sure you have scooped out all the dark black membranes underneath each cap.
In a small bowl, mix together the vinegar, garlic, anchovies, marjoram, and olive oil. Pour this over radicchio and portobellos and let stand for 1 hour.
Preheat the broiler.
Place the portobellos and radicchio on a cookie sheet, reserving the marinade that drips off, and broil until just lightly charred on one side, 4 to 5 minutes. Turn them over and broil for 3 to 4 minutes on the other side. Remove, and set aside until they are cool enough to handle.
Cut the mushrooms into ¼-inch-thick strips, and cut the radicchio in half lengthwise. In a large bowl, combine the mushrooms, radicchio, and the reserved marinade. Toss gently to thoroughly coat ingredients with the marinade.
Spoon the polenta on a large platter, arrange the mushroom-radicchio mixture in the center and serve.