Season and prep the chicken. Pat the chicken dry with paper towels. Season generously with salt and pepper on both sides. Zest one lemon and set the zest aside.
Cut another lemon into wedges for serving.
Brown the chicken. Heat 1 tablespoon olive oil in a large, wide skillet or Dutch oven over medium-high heat. Add the chicken skin-side down and cook until deeply golden and crisp, about 6–8 minutes. Flip and brown the other side for 3–4 minutes.
Transfer the chicken to a plate. Don’t wipe the pan—those browned bits are flavor.
Soften the aromatics. Lower heat to medium. Add 1 tablespoon butter to the pan.
Stir in the diced onion and a pinch of salt. Cook until soft and translucent, 3–4 minutes. Add minced garlic and cook 30 seconds until fragrant.
Toast the rice. Stir in 1.5 cups long-grain rice.
Cook, stirring, for 1–2 minutes to coat the grains in fat and pick up the browned bits. This step helps keep the rice fluffy.
Deglaze and season. Pour in 1/2 cup chicken stock and scrape the bottom of the pan to lift any stuck-on bits. Add the remaining 2 cups stock, lemon zest, 2–3 tablespoons fresh lemon juice, a few thyme sprigs, and a pinch of red pepper flakes if using.
Taste the liquid and adjust salt and lemon now; it should be well-seasoned and bright.
Nestle the chicken back in. Return the chicken to the pan, skin-side up, along with any juices. Bring to a gentle simmer, then reduce heat to low, cover, and cook for 18–22 minutes. The rice should be tender and the chicken cooked through (165°F/74°C).
Finish with butter and herbs. Remove the lid and take out the thyme sprigs.
Stir 1 tablespoon butter into the rice for a silky finish. Sprinkle in chopped parsley. If using capers, add 1–2 tablespoons now for a briny contrast.
Rest and serve. Let the pan sit off the heat for 5 minutes to settle.
Serve with lemon wedges for squeezing at the table. The extra hit of fresh lemon right before eating makes the flavors pop.