Place a large Dutch oven over medium-high heat. Place the chicken pieces onto a paper towel and pat dry to help create extra-crispy skin. Season the chicken thighs on both sides with kosher salt and freshly ground black pepper. Add the blended oil to the pot and add the chicken thighs, skin-side down and allow to sear until golden brown, about 3 to 5 minutes per side. Flip and continue to brown moving the thighs to the side of the pan. To the other side of the pan, add the shallots, garlic, sliced Fresno chiles, salt, bay leaves, cinnamon sticks, coriander seeds, cumin seeds, and stir to combine. Add the fingerling potatoes and season with kosher salt.
Add about 2 cups chicken stock or enough to come 2/3 of the way up the meat and add the whole peeled San Marzano tomatoes, crushed by hand, along with half the juices/sauce. Stir in the cerignola olives, oil-cured olives, and kalamata olives. Taste the mixture and season with salt and pepper, if needed. Lower the heat to medium and cover with a lid. Cook for about 20 to 30 minutes, stirring occasionally.
When ready to serve, add the orange zest and stir to combine.
For the salad: Place the shallots, vinegar, and salt in a medium bowl. Zest the orange. Using a sharp knife, remove the peel and pith from the orange. Holding the orange over a bowl (to catch any juice), supreme or segment the orange. Set the supremes aside. Squeeze the remaining orange membrane into the bowl to collect all the orange juice. Whisk in about 1/2 cup extra-virgin olive oil. Taste and season with salt, if needed.
To another medium bowl, toss in 1 small bunch fresh basil leaves and a small bunch arugula leaves. Add a small amount of dressing and toss to combine. For serving: Plate the chicken pieces onto a serving platter. Top with the potatoes, olives, and tomatoes. Season the chicken mixture with sea salt and top with the dressed greens. Serve with remaining sauce alongside the table.
Rinse the chicken parts thoroughly. In a 10-quart stockpot, combine the onion, carrot, garlic, thyme, salt, bay leaf, peppercorns, and 1 gallon cold water. Bring the liquid to a boil over high heat, skimming and discarding any foam and impurities that rise to the surface. Reduce the heat to low and simmer 5 hours, skimming the surface as necessary. Strain the stock through a fine-mesh strainer, discarding the solids.
Makes 3 quarts