Heat the oven to 375ºF and set a rack in the middle slot.
Dry the chicken well and trim any excess fat. Season both sides of the chicken generously with kosher salt and pepper. Heat the oil in a large (12-inch), oven-safe skillet on medium-high. When the oil is shimmering, add the chicken in a single layer, skin-side down. Flatten each piece to ensure good contact with the pan. Let it cook, undisturbed, until the skin is crisp and deep golden brown, about 10 to 12 minutes. Remove the chicken to a plate, skin-side up and set it aside (the chicken will not be fully cooked at this point).
If there is a lot of fat in the pan, remove all but 2 tablespoons. Turn the heat down to medium, add the shallots and cook, stirring often, until they turn soft and translucent, about 5 minutes. Add the fresh garlic and black garlic and cook until fragrant, about 1 minute. Sprinkle in the flour and stir until no dry flour remains, about 1 minute. Stir in the wine, cream and chicken broth and bring to a simmer, stirring often. Add the chicken back into the pan, skin-side up, place in the oven and bake until the chicken is fully cooked through (165ºF on an instant read thermometer), about 15-30 minutes, depending on the size of the pieces (breasts will cook faster so check after 10 minutes).
Remove the skillet from the oven, transfer the chicken to a serving platter and cover it loosely with foil to keep warm. Turn the heat under the skillet on low, and stir in the spinach, a handful at a time, until it’s wilted but still bright green, about 3 minutes. Taste the sauce and season it with salt and pepper, as desired. Spoon the sauce around the chicken and sprinkle with chopped parsley and squirt some lemon juice on top just before serving.