Preheat oven to 425°F. Pat the chicken thighs dry with paper towels. In a small bowl whisk 2 tbsp olive oil, lemon juice, minced garlic, dried oregano, and 0.75 tsp salt and 0.5 tsp pepper. Rub the mixture all over the thighs and let rest at room temperature for 10 minutes while you prep the vegetables.

In a large bowl toss the halved potatoes, carrots, and Brussels sprouts with the remaining 1 tbsp olive oil, 0.75 tsp salt, and 0.5 tsp pepper so everything is evenly coated. Spread them in a single layer on a rimmed baking sheet, cut side down for potatoes and sprouts.

Roast the vegetables in the preheated oven for 10 minutes to give them a head start; they should begin to brown on the edges.

While the vegetables roast, heat an ovenproof skillet over medium-high heat. Add the skillet to the stove with the remaining 1 tbsp olive oil (if needed) and when shimmering, place the chicken thighs skin-side down. Sear without moving for 5–7 minutes until the skin is deep golden and fat has rendered. Flip and sear the other side 2–3 minutes just to brown.

Transfer the seared thighs to the baking sheet, skin-side up, nestling them among the partially roasted vegetables. Return the sheet to the oven and roast 12–15 minutes more, until a meat thermometer inserted into the thickest part reads 165°F and potatoes are tender when pierced.

Remove the chicken and vegetables to a platter and tent loosely with foil to rest. Place the skillet used for searing over medium heat (careful if it's hot) and pour in the chicken broth to deglaze, scraping up browned bits with a wooden spoon. Let simmer 1–2 minutes, then whisk in the cold butter a tablespoon at a time until the sauce is glossy. Taste and season with more salt or a squeeze of lemon if needed.

Spoon the lemon-butter sauce over the chicken and roasted vegetables, sprinkle with chopped parsley, and serve with lemon wedges.