Preheat the oven to 425°F (220°C). This high heat will help render the chicken skin fat quickly and crisp the skin while roasting the potatoes to a golden exterior.

Pat the chicken thighs very dry with paper towels and season both sides generously with kosher salt and a few grinds of black pepper (and the smoked paprika if using). Dry skin promotes better browning and a crisp crust; salt early on the skin also draws out a bit of surface moisture so the skin sears rather than steams.

Cut potatoes into even 1-inch pieces so they cook uniformly. Toss potatoes with 1 tablespoon olive oil, 1/2 teaspoon kosher salt and a few grinds of black pepper. A light coat of oil helps conduct heat and creates a crunchy, caramelized surface.

Heat a large, oven-safe skillet (preferably cast iron) over medium-high heat until hot, then add 1 tablespoon olive oil. Add the chicken thighs skin-side down and sear without moving for 4–5 minutes, until the skin is deep golden-brown. Searing develops the Maillard reaction — complex flavors and color that make the dish taste roasted, even before the oven.

Flip the thighs and sear the other side 1–2 minutes just to color the meat. Remove the chicken to a plate and pour off and reserve any excess rendered fat if there’s more than 2 tablespoons — too much fat will make the potatoes soggy.

In the same skillet, add the remaining 1 tablespoon olive oil and the potatoes in a single layer, cut-side down where possible. Let them cook undisturbed over medium-high heat 5–7 minutes until the undersides are deeply browned. Browning builds flavor and gives crispness that contrasts the tender interior.

Nestle the seared chicken thighs skin-side up among the potatoes. Transfer the skillet to the preheated oven and roast 20–25 minutes, until an instant-read thermometer inserted into the thickest part of a thigh reads 165°F (74°C) and the potatoes are tender. Finishing in the oven ensures even cooking without burning the skin.

While the chicken roasts, melt 4 tablespoons butter in a small saucepan over medium-low heat. Add the minced garlic and cook gently 30–45 seconds — just until fragrant — to avoid bitterness. Stir in the lemon zest, lemon juice, chopped parsley and thyme leaves. The butter acts as a carrier to bloom the herbs and garlic and will become the finishing sauce that brightens the dish.

When the chicken and potatoes come out of the oven, transfer the chicken to a cutting board and tent loosely with foil to rest 5 minutes (resting redistributes juices). Spoon 2 tablespoons of the lemon-garlic butter over each thigh to melt into the skin and keep the meat glossy.

While the meat rests, bring the skillet with pan juices back to medium heat on the stove. Add the chicken broth and scrape the bottom of the pan with a wooden spoon to release browned bits (fond). Reduce 2–3 minutes until slightly thickened; this creates a quick pan sauce that captures the roasted flavors. Turn off the heat and whisk in 1 tablespoon of the prepared lemon-herb butter to finish and round the sauce.

Quickly blanch the green beans in a pot of salted boiling water 2 minutes, then shock in an ice bath to stop cooking (this preserves bright color and crisp-tender texture). Drain. In a skillet, melt 1 tablespoon butter with a splash of olive oil, add the beans, a pinch of salt and sauté 2–3 minutes until warmed through and glossy. Finish with a squeeze of lemon if desired.

To serve: divide potatoes among 4 plates, place a chicken thigh on each, spoon the pan sauce over the meat and potatoes, and arrange the green beans alongside. Garnish with extra parsley and a lemon wedge. Resting the chicken and finishing with a hot sauce keeps the skin crisp and the meat juicy.