Preheat oven to 350 degrees F and set the oven rack into the middle of the oven.
Season the chicken with some salt and pepper on both sides.
Place a cast iron pan, 9 or 10 inches large, over high heat and once hot add the oil.
When the oil is hot, add the chicken, (skin side down, if using skin-on chicken thighs) and cook until browned, 4-5 minutes on each side. Set aside onto a plate.
To the same pan, add the garlic and cook stirring for 1 minute.
Add the rest of the sauce ingredients and cook on medium heat until the sauce reduces in volume and thickens slightly (about 5-8 minutes). Stir occasionally.
If you want the sauce thicker, cook on low for longer, or add the mentioned cornstarch slurry by mixing two tablespoons of cornstarch and water until fully dissolved and adding the mixture to the sauce.
Transfer sauce to a bowl. Set aside.
Reduce heat to low-medium and add butter to the same cast iron skillet.
Once the butter is melted, add the diced onion and cook for 1 minute, stirring.
Add in the uncooked rice, dried rosemary, and lemon zest. Stir and cook for a minute.
Pour in the chicken broth, stir to combine.
Add the meat back to the pan, on top of the rice. If using skin-on thighs, place them skin side up.
Cover with foil and bake in the preheated oven for 45-50 minutes. When 5 minutes are left, uncover, brush the meat with the sauce and broil for 5 minutes.
Serve with lemon slices, fresh parsley, and extra Honey Lemon Sauce.