Preheat the oven to 400 degrees F. Cut a piece of aluminium foil about 12 inches long.

Put the garlic on 1/2 of the foil, sprinkle with salt and pepper, drizzle in some olive oil, and add 2 sprigs of thyme. Fold the foil enclosing the garlic, making a pouch, and add a couple teaspoons of water. Fold the foil over the garlic to enclose and fold in the edges 2 or 3 times to seal. Put the package in the oven and roast for 30 minutes, or longer, until the garlic is soft. Open the package and let the garlic cool a bit, then squeeze out the cloves into a food processor or blender. Add 1/2 cup olive oil, the lemon juice, parsley, and thyme leaves and puree to make a thick vinaigrette.

Preheat an outdoor gas or charcoal grill to medium heat.

Rinse the chicken and pat dry with paper towels. Put the pieces in a bowl, sprinkle with salt and pepper, and drizzle with olive oil; toss to coat with the seasonings and then refrigerate while you get everything else together.

Put the radicchio in another bowl, drizzle with olive oil, balsamic vinegar. Sprinkle with salt and pepper and toss; set aside.

When you're ready to cook, take a few paper towels and fold them several times to make a thick square. Blot a small amount of oil on the paper towel and carefully and quickly wipe the hot grates of the grill to make a nonstick surface.

Season with salt and pepper and place the radicchio quarters on the preheated grill. Cook the radicchio for about 2 minutes per side until it has a nice char; pull it off the grill, place into a bowl, cover with plastic wrap, and set aside to steam and finish cooking.

Then put the chicken on the grill, skin side down, and grill. Grill the chicken for about 20 minutes, turning once, then baste with about half of the vinaigrette and keep cooking until an instant-read thermometer stuck into the thickest part of the thigh reads 160 degrees F and the chicken is nice and caramelised all over, 15 to 20 more minutes. During the last few minutes, throw the lemon halves on the grill, cut sides down, and cook until just marked and smoky.

To serve, separate the radicchio into individual leaves in a big bowl. Add the chicken and the rest of the dressing and toss well. Serve with grilled lemon halves, squeeze the lemon over the chicken, and top with reserved dressing.