Pat the chicken dry and, using a meat mallet or rolling pin, gently pound to an even thickness (about 3/4 inch). Season both sides with 1 tsp kosher salt, 1/4 tsp black pepper, smoked paprika, and half the minced garlic. Toss with 1 tbsp olive oil and the lemon zest; set aside at room temperature while you preheat the oven to 425°F.

On a rimmed baking sheet, toss the halved potatoes with 1 tbsp olive oil, 1/2 tsp kosher salt, 1/4 tsp black pepper, and the thyme. Spread cut-side down and roast at 425°F for 20 minutes. Remove the sheet, add the broccoli florets tossed with a pinch of salt and a drizzle of oil, toss to combine, and return to the oven for 8–10 more minutes until potatoes are golden and broccoli is tender with some charred edges.

While the vegetables roast, heat a large skillet over medium-high heat until shimmering. Add 1 tbsp olive oil and sear the chicken 4–6 minutes on the first side without moving it, until deeply golden. Flip and cook another 4–6 minutes, then reduce heat to medium and add the remaining minced garlic and butter to the pan. Spoon melted butter over the chicken for 30–60 seconds. Cook until internal temperature reaches 165°F (about 6–12 minutes total depending on thickness). Transfer chicken to a plate and tent with foil to rest 5 minutes.

With the skillet still over medium heat, pour in the lemon juice and scrape up any browned bits with a wooden spoon. Let simmer 1–2 minutes, then swirl in the remaining 1 tbsp butter (if desired) to gloss the sauce. Taste and adjust seasoning with remaining salt and pepper. Return chicken to the pan for 30 seconds to coat in the sauce.

Divide roasted potatoes and broccoli among 4 plates, place a chicken breast on each, spoon extra pan sauce over the chicken, and sprinkle with chopped parsley. Serve immediately with extra lemon wedges if desired.