Preheat the oven to 375F.
Pat the chicken dry, then season your chicken pieces on all sides with 1 teaspoon of salt and some grated black pepper.
In a large oven-proof skillet (I used a 12-inch cast iron), warm the olive oil over medium heat. Place the chicken into the skillet skin-side down. Gently press with a spatula to make sure the skin is in contact with the pan. Once it starts sizzling, let it cook, undisturbed, for 4-5 minutes or until the skin is light golden brown (it will continue to brown in the oven).
The skin should easily release from the pan when you flip it. If it’s sticking, let it keep cooking. Flip and cook for 1 minute, then transfer to a plate.
To the skillet over medium heat, add the onion and ½ teaspoon salt. There should be a tablespoon or two of oil/fat in the pan - if the skillet looks dry, add olive oil. Cook the onions for 3-5 minutes or until softened, stirring occasionally.
Add the potatoes, brussels sprouts, and remaining ½ teaspoon salt. Toss everything together so the potatoes and sprouts get coated in the oil and salt.
In a bowl or liquid measuring cup, add the water, mustard, and vinegar and use a fork to mix it all together. Pour it into the skillet, then turn off the heat. Nestle the rosemary sprigs into the liquid/veg. Arrange the chicken on top (skin-side-up).
Bake for 35-40 minutes or until the liquid has reduced and the chicken skin is a deep golden brown color. Let it rest for a few minutes to allow things to cool down and the liquid to thicken.
Place the chicken on your plate, scoop the potatoes/veg alongside. Use a spoon to scoop any liquid at the bottom of the pan and drizzle it all over everything. Top with a generous squeeze of lemon and some parsley, if using.