Garlicky Chicken With Herbs & Vinegar

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3 pounds bone-in, skin-on chicken legs and thighs
Kosher salt and freshly ground black pepper
2 ounces Pecorino Romano or Parmesan (or a scant 1/2 cup pre-grated)
5 large garlic cloves
5 oil-packed anchovy fillets
2 tablespoons dried oregano
2 tablespoons dried thyme
1/2-1 teaspoons red pepper flakes
4 tablespoons olive oil, divided
3/4 cup dry white wine, chicken broth, or a mix
2 tablespoons red wine vinegar, divided
2 tablespoons balsamic vinegar, divided
Small handful fresh oregano and/or thyme sprigs
Freshly chopped parsley, for serving
Good Italian bread, for serving

Heat the oven to 450ºF. Pat-dry the chicken and season well on all sides with salt and pepper.
If not using pre-grated, cut the cheese into a couple pieces, place in a food processor and pulse until the cheese is finely grated (or, if using pre-grated, just add the cheese to the machine). Add the garlic, anchovies if using, herbs, red pepper flakes, and 3 tablespoons of olive oil. Pulse until a paste forms. (This mixture can also be made by hand: Grate the cheese on the fine holes of a box grater. Finely chop the herbs and garlic before mixing with red pepper flakes and oil.) Season with a big pinch of salt and lots of pepper.
Heat remaining tablespoon oil in a large oven-safe (stainless steel or cast-iron work best) skillet over medium-high heat. Nestle in half the chicken parts, skin-side down. Sear the chicken until it releases from the pan naturally and is well-browned, 8 to 10 minutes. Flip the chicken and let cook for another 8 minutes. Transfer the chicken to a sheet pan (or plate or platter) skin-side up, then repeat with remaining chicken parts.
Use a spoon or spatula to smear the cheese-herb mixture over the skin side of each chicken piece.
Pour the white wine or broth, and 1 tablespoon each of the balsamic and red wine vinegars into the skillet, and use a wooden spoon to loosen any browned bits stuck to the pan. Scatter sprigs of fresh oregano and thyme, then return all the chicken, still skin-side up, to the skillet. Transfer the skillet to the oven and cook until the liquid has reduced by about half and the chicken is cooked through, registering at least 165ºF on an instant-read thermometer, about 15 minutes. Transfer the chicken to a serving platter deep enough to hold the sauce.
Scrape up any browned bits in the pan and stir in the remaining tablespoon of both vinegars. Pour pan sauce over the chicken and serve topped with parsley.