Add the chicken, buttermilk, 2 sprigs of rosemary, 1 teaspoon salt and 1/2 teaspoon pepper to a large resealable plastic bag. Let marinate, refrigerated, for 4 hours or up to overnight.
Preheat the oven to 425 degrees F.
Add the potatoes, 2 tablespoons chopped rosemary, 2 teaspoons salt, 1 teaspoon pepper and 2 tablespoons olive oil to a large bowl. Toss together until everything is coated. Pour onto a rimmed baking sheet and spread in an even layer. Reserve the bowl. Bake for 40 minutes.
Add the broccoli to the same bowl. Add 2 tablespoons olive oil, 1/2 teaspoon salt and a little lemon zest. The Add broccoli to the baking sheet with the potatoes and toss to combine. Return to the oven and cook until tender, about 20 minutes.
Meanwhile, heat a large cast-iron skillet over medium-high heat. Add the remaining 2 tablespoons olive oil. Remove the chicken thighs from the marinade one by one, shaking the remaining buttermilk off lightly, and add to the skillet. Cook until the thighs are slightly charred, 6 to 7 minutes. Flip and cook on the other side until fully cooked, another 6 to 7 minutes. Serve the chicken with a side of potatoes and broccoli. Garnish with fresh rosemary and lemon juice.