Place oven rack at lowest level and preheat oven to 450.
Toss bread cubes with 1 tablespoon of the oil, 1/4 teaspoon salt and pepper to taste. Place bread in 8 inch baking dish and set aside.
Dry chicken with paper towels and season with salt and pepper.
Heat 1 tablespoon of the oil until hot. Place chicken in skillet, skin side down, and saute until skin is deep golden color, about 5 minutes.
Place chicken, skin side up, on bread cubes in baking dish and bake until the thickest part of the breast is 160-165 degrees and the bread is spotty brown, about 18 to 20 minutes. Take care, don't overcook the breasts!
Transfer chicken to cutting board. When chicken is cool enough, take meat off bones and cut into pieces similar in size to bread cubes. Throw away the bones.
While chicken is baking, pour off fat in skillet, add the last tablespoon of oil, the garlic and thyme, and cook for a moment until the garlic is fragrant. Add broth and scrape up any brown bits from skillet. Cook until thickened slightly, stir in vinegar, cover, and take off the heat.
Toss bread cubes in a bowl with most of dressing mixture in skillet - reserve 2 tablespoons - add chicken and any juices from it. Then add arugula, tomato, pine nuts and the remaining dressing. Season with salt and pepper to taste - remember, the cheese will add some salty taste. Toss a final time. Garnish with Parmesan shavings and serve.
(To gild the lilly a bit, you can add what Lydia Bastianich calls a final "blessing" of extra virgin olive oil. Or, put that on the table. Some will find it fine without.).