Cut 1 1/2 inches from the root end of the 2 endives and the radicchio head, and cut these root end pieces into 1/2-inch chunks. (You should have 3 about cups.) Reserve the pointed endive leaf tips and the radicchio leaves for use later in the recipe. Combine the root end chunks with the shredded carrot and nuts in a bowl.
Combine the dressing ingredients in a small bowl, then add the dressing to the endive and radicchio pieces, carrots, and nuts. Toss well.
Arrange the radicchio leaves attractively on four plates, and spoon the dressed mixture into the center of the leaves. Arrange the endive leaf tips so they stand, pointed tips up, next to one another all around the dressed salad. Serve immediately.