Preheat the oven to 400 degrees F.
To prepare the vinaigrette: Combine the egg yolks, soy sauce, vinegar, mustard, and honey in a blender. Blend on medium speed until thoroughly combined. With the blender running, slowly add the peanut and sesame oils, until emulsified. Taste for seasoning, adjust with salt and pepper. Set aside.
Season the chicken on both sides with salt and pepper. Place a large ovenproof saute pan over medium heat and coat with the oil. When the oil is very hot and nearly smoking, lay the chicken breasts in the pan, skin-side down. Cook until the skin is golden brown, about 5 minutes. Transfer the pan to the oven and continue to cook for another 10 minutes or until the chicken is firm to the touch. Remove from the oven and set aside on a plate to cool.
Separate a few leaves from the Napa cabbage and the radicchio for lining the serving plate. Finely chop the remaining Napa cabbage and radicchio heads and put in a large salad bowl. Add the mixed greens. Drizzle enough of the vinaigrette onto the lettuces and toss gently with your hands until well coated. Season with salt and pepper.
Remove the skin from cooled chicken. Put the chicken on a cutting board and slice into thin strips. Add the chicken to the bowl of salad greens. Toss gently with your hands to distribute. Season with salt and pepper.
Slice the wonton skins into 1/4 inch ribbons. Heat 3 inches of canola oil in a wok to 350 degrees, and check with a frying thermometer to ensure accurate temperature. Carefully add the sliced wontons to the oil and fry for 1 to 2 minutes until golden and crisp. Using a mesh strainer or slotted spoon, remove the fried wontons to a plate lined with paper towels to drain.
Arrange the whole Napa cabbage and radicchio leaves nicely on a serving platter. Put a large mound of the salad on top, followed by the crisp wontons. Garnish with a crown of watercress, scallions, and sesame seeds.