Tear off the large tough stems from the bottom of the arugula. Wash and dry the arugula leaves. Cut the leaves into 1/2-inch shreds. Put the shredded arugula in a large bowl.
Pull off 20 or so of the largest, nicest outer leaves from the endive. They are going to be your endive cups. Arrange them around the center of a large platter (or 2 smaller plates), leaving space in the center. Cut the remaining endive crosswise into 1/4-inch shreds, starting at the pointy top and stopping when you get to the bitter core. Put the shredded endive in the bowl with the arugula.
Peel any outer wilted leaves from the radicchio and cut the head in half. Cut out the core and then cut the radicchio into 1/4-inch shreds. Add to the bowl with the arugula and the endive. (You can make the endive cups and prepare the greens up to a couple hours in advance. Cover them with damp paper towels and keep them in the refrigerator until you're ready to dress the salad.)
To make the dressing: place all the dressing ingredients in a sealable container and shake until totally blended.
Just before serving, toss the salad with most of the dressing. Fill the endive cups with salad and splatter with the remaining dressing. Serve immediately.