Season the chicken: Use paper towels to pat the chicken pieces dry. Season well with kosher salt and black pepper on both sides. Rub some salt and pepper underneath the skin as well.
Sear the chicken: In a large nonstick skillet set over medium-high heat, add 3 tablespoons of olive oil. Once the oil begins to shimmer, add the chicken, skin side down. Cook until the skin is crispy and golden brown, about 5-8 minutes. Flip and cook until the second side is golden brown as well, about 5 minutes more.
Season and rest the chicken: Transfer the chicken to a large tray and sprinkle with Italian seasoning. Pour the lemon juice over the chicken. Set it aside for now.
Start the sauce: Turn the heat to medium. If the skillet is dry, add a little more olive oil. Add the anchovy filets. Cook for about 2 minutes or so, pushing the anchovies around the skillet. They will kind of melt into the oil and infuse it with flavor.
Finish the sauce: Add the garlic and cook until fragrant, about 30 seconds. Add the tomatoes, half of the olives, half of the capers, and the dried oregano and red pepper flakes (if using). Bring the sauce to a simmer over medium heat.
Cook the chicken in the sauce: Add the chicken back into the skillet, nestling it in the sauce. Spoon a bit of the sauce over the chicken. Cover part way, either with a lid or a splatter guard. Cook until the chicken is fully cooked and tender, about 25-30 minutes, depending on how large the pieces are. A thermometer in the thickest part of the chicken should read 165°F.
Serve: Garnish with the parsley and the remaining olives and capers and serve.
If you have a smaller skillet, brown the chicken in batches to avoid overcrowding the pan. This will help crisp the skin.
Let the chicken sit, skin-side down, until the skin releases from the pan and becomes deeply golden brown. This takes about 5-8 minutes, and is worth every second.