Pre-heat oven to 375°F. Place chicken pieces atop a rimmed baking sheet in a single layer and season generously with salt and pepper all over and on both sides.
Over medium-high heat in a large Dutch oven, bring 2 tablespoons of the olive oil to shimmering. Beginning with the thigh pieces, place into the hot oil and sear undisturbed for 2 minutes. Turn the pieces, searing on the second side for 1 minute more and golden brown. Use a slotted spoon to remove the thigh pieces to a bowl and cover tightly with foil before repeating the process with the breast pieces. NOTE: the chicken will NOT be cooked through, rather will finish cooking during the braising step.
Add remaining tablespoon oil to the Dutch oven. Add in diced onion and sauté until translucent, 3 to 4 minutes. Add garlic and cook one minute more.
Tie off and secure the thyme branches, sprigs of parsley, and single bay leaf into a bouquet garni. Add gem peppers, carrot, dried oregano, and the bouquet garni to the Dutch oven and stir well to combine. Reduce the heat to medium-low and cook until the veg are soft, about 5 minutes. I like to tie the end of my bouquet garni string to the handle of my Dutch oven, so I am not left fishing it out later.
Deglaze the pot wine using a wooden spoon to scrape up the browned bits. Cook until the wine has reduced significantly, at least 2 minutes. Make a well in the center of the softened vegetables and add in tomato paste. Cook the paste for several minutes until it turns brown.
Add in crushed tomatoes, torn fresh basil leaves, and red pepper flakes and stir. Return the chicken pieces along with all accumulated juices to the pot. Season with salt and pepper.
Add the lid onto the pot leaving the bouquet garni string tied to the handle and the herbs submerged in the sauce. Place the covered Dutch oven into the pre-heated oven and braise for 35 minutes.
After 35 minutes, remove the lid, add the olives, and place back into the oven, uncovered, for another 10 to 12 minutes. Remove from the oven and let stand for 10 minutes before dusting with parmesan cheese and serving.