Step 1Cut the radicchio into quarters through the root end. Brush the cut side of the radicchio with 1 tablespoon olive oil.
Step 2Grill over medium heat, cut side down, for 3 to 6 minutes, until lightly charred on the outside but not cooked through. The center of the radicchio should be uncooked. Remove from heat.
Step 3Slice the radicchio crosswise into thin strips. Toss with the red onion.
Step 4Whisk the balsamic vinegar, mustard and honey in a small saucepan. Bring the mixture to a boil over medium-high heat, and cook, at a medium boil, for 1 minute. Remove from heat. Whisk in remaining olive oil. Season to taste with salt and pepper.
Step 5Pour the vinaigrette over the radicchio and onion. Season to taste with salt and pepper. This can be made several hours ahead and refrigerated until ready to serve. Makes 6 cups.