Adjust oven rack to lower-middle position; heat oven to 300 degrees. Sprinkle both sides of chicken with salt. Heat 1 teaspoon oil in Dutch oven over medium-high heat until shimmering but not smoking. Add 4 chicken thighs, skin-side down, and cook without moving them until skin is crisp and well browned, about 5 minutes. Using tongs, turn chicken pieces and brown on second side, about 5 minutes longer; transfer to large plate. Add remaining 4 chicken thighs to pot and repeat, then transfer to plate and set aside. Discard all but 1 tablespoon fat from pot.
Add onion to fat in Dutch oven and cook, stirring occasionally, over medium heat until browned, about 4 minutes. Add garlic, anchovy, and cayenne; cook, stirring constantly, until fragrant, about 1 minute. Add wine and saffron and scrape up browned bits from pan bottom with wooden spoon. Stir in chicken broth, tomatoes, tomato paste, thyme, oregano, bay, and herbes de Provence (if using). Remove and discard skin from chicken thighs, then submerge chicken in liquid and add accumulated chicken juices to pot. Increase heat to high, bring to simmer, cover, then set pot in oven; cook until chicken offers no resistance when poked with tip of paring knife but still clings to bones, about 1 1/4 hours.
Using slotted spoon, transfer chicken to serving platter and tent with foil. Discard bay leaf. Set Dutch oven over high heat, stir in 1 teaspoon orange zest, bring to boil, and cook, stirring occasionally, until slightly thickened and reduced to 2 cups, about 5 minutes. Stir in olives and cook until heated through, about 1 minute. Meanwhile, mix remaining 1/2 teaspoon zest with basil. Spoon sauce over chicken, drizzle chicken with remaining 2 teaspoons olive oil, sprinkle with parsley mixture, and serve.
Tip: Removing the pits from tiny niçoise olives by hand is not an easy job. Cover a cutting board with a clean kitchen towel and spread the olives on top, spacing them about 1 inch apart. Place a second clean towel over the olives. Using a mallet, pound all of the olives firmly for 10 to 15 seconds, being careful not to split the pits. Remove the top towel and, using your fingers, press the pit out of each olive.