Preheat oven to 425°F. Pat the chicken thighs dry with paper towels (dry skin crisps better). In a small bowl, combine lemon zest, lemon juice, minced garlic, 2 tbsp olive oil, thyme, smoked paprika, 1 tsp kosher salt, and 1/4 tsp black pepper.

Toss the halved potatoes in a mixing bowl with the remaining 1 tbsp olive oil, 1/4 tsp kosher salt, and the remaining 1/4 tsp black pepper. Spread potatoes in an even layer in the skillet (or on a sheet pan if your skillet is small).

Nestle the chicken thighs skin-side up among the potatoes. Rub the lemon-garlic mixture evenly over the chicken and drizzle any leftover over the potatoes. Place the skillet on the stovetop over medium-high heat and sear for 4–6 minutes just to render some fat and start browning the skin (you should see the edges turn golden).

Transfer the skillet to the oven and roast at 425°F for 25–30 minutes, until the potatoes are tender when pierced and the chicken reaches 165°F at the thickest part (skin should be deep golden). If skin isn’t as crisp as you like, broil on high for 1–2 minutes while watching closely.

Remove chicken and potatoes to a platter to rest. Place the skillet over medium heat, add the chicken broth and the butter, and scrape up any browned bits with a wooden spoon. Simmer for 2–3 minutes until slightly reduced, then spoon the sauce over the chicken and potatoes.

Let the chicken rest 5 minutes before serving so juices redistribute. Serve hot with lemon wedges and an extra sprinkle of fresh thyme if desired.