Preheat and prep: Heat the oven to 425°F (220°C). Line a large rimmed sheet pan with parchment or foil for easy cleanup. Pat the chicken dry with paper towels to help the skin crisp.
Season the potatoes: In a bowl, toss the halved baby potatoes with 1 tablespoon olive oil, half the minced garlic, 1 teaspoon rosemary, half the thyme, 1/2 teaspoon salt, and a pinch of pepper.
Spread the potatoes cut-side down on the pan. Add a few rosemary sprigs if you have them.
Make the chicken rub: In a small bowl, combine 1.5 tablespoons olive oil, the remaining garlic, remaining rosemary and thyme, smoked paprika, lemon zest, 1 teaspoon salt, and black pepper. Rub this mixture under the skin and over the outside of the chicken pieces.
Arrange on the pan: Nestle the chicken thighs among the potatoes, skin side up, without crowding.
This helps everything brown nicely.
Roast the first round: Place the pan on the middle rack and roast for 25 minutes. The potatoes should be starting to color and the chicken skin should be rendering.
Add the green beans: Toss the green beans with the remaining 1/2 tablespoon olive oil and a light sprinkle of salt. Pull the pan out, scatter the green beans around the chicken, and drizzle over the balsamic or honey if using.
Return to the oven.
Finish roasting: Roast another 12–18 minutes, until the chicken reads 165°F (74°C) at the thickest part and the potatoes are tender. If you want extra-crisp skin, broil for 1–2 minutes at the end, watching closely.
Optional butter baste: Dot the hot pan with small pieces of butter and let it melt into the juices. Spoon the pan juices over the chicken and vegetables.
Rest and serve: Let the chicken rest 5 minutes.
Squeeze lemon wedges over everything for brightness. Serve right from the pan.