Make the mustard vinaigrette: In a bowl, combine mustard, tarragon, sugar, vinegar, salt, and pepper and whisk well. While whisking, slowly drizzle in the oil. Drizzle until the oil is incorporated into the dressing. Season, to taste, with salt and pepper. Set aside.
Preheat a grill or grill pan.
Bring a small saucepan of water to a boil. Add the whole garlic cloves and blanch for 1 1/2 minutes. Drain and thinly slice. Set aside.
Using your fingers, gently separate the skin from the chicken breasts, making a pocket. Be sure to keep the skin attached to the chicken breast. Place a few blanched garlic slices and parsley leaves under the skin of the chicken breast. Pull the skin back over the breasts. Season the outside well with salt and pepper. Drizzle the breasts with 3 tablespoons of the olive oil.
Place the chicken breasts on the grill, skin-side down, and cook until golden brown, about 6 minutes. Turn the chicken and cook until cooked through and the juices run clear, about another 6 to 8 minutes depending on the thickness of the chicken breasts. Remove and set aside.
Heat a saute pan over medium-high heat. Add the remaining 3 tablespoons olive oil and heat. Add the shallots and the chopped garlic and saute 1 minute. Add the mushrooms and saute until golden brown and cooked, about 8 minutes. Remove from the heat and set aside.
Slice each chicken breast into 4 slices on the diagonal.
Combine the watercress and the chopped radicchio in a large bowl. Toss with desired amount of dressing. Place the whole radicchio leaves around the edges of 6 plates. Pile the watercress mixture in the center. Divide the mushrooms among the plates. Top each plate with the sliced chicken breast. Drizzle with a little more dressing.