Preheat and prep: Position a rack in the middle of the oven and preheat to 425°F (220°C). High initial heat ensures quick browning of the chicken skin and caramelization of the vegetables—this develops flavor and creates a crisp exterior.

Season the chicken: Pat the chicken thighs very dry with paper towels (moisture is the enemy of crisp skin). In a bowl, whisk 2 tbsp olive oil, minced garlic, chopped rosemary, thyme, lemon zest and juice, Dijon, 1 tsp kosher salt and 1/2 tsp black pepper. Rub this mixture under the skin and over the skin of each thigh and refrigerate uncovered if you have 15–30 minutes (drying the skin briefly helps it crisp in the oven).

Parboil the potatoes: While the chicken marinates, put potatoes in a large pot, cover with cold water by 1" and add 1 tsp kosher salt. Bring to a simmer and cook until a fork slides through easily, about 12–15 minutes. Starting in cold water brings the interior and exterior to temperature together, so the potatoes cook evenly and mash silkily.

Roast the chicken: Arrange marinated thighs skin-side up on a rimmed baking sheet or in a large roasting pan with a little space between pieces (crowding will steam the skin). Roast at 425°F (220°C) for 35–40 minutes until the skin is golden and an instant-read thermometer inserted into the thickest part away from bone reads 165°F (74°C). If the skin needs extra browning, broil for 1–2 minutes while watching closely. Roasting at high heat renders fat quickly and crisps skin without drying the meat.

Start the green beans: Toss trimmed green beans with 1 tbsp olive oil, sliced garlic, a pinch of salt and pepper and spread on a separate rimmed baking sheet. Roast in the same 425°F oven for 12–15 minutes (they take less time than the chicken). Roasting concentrates natural sugars for a deeper flavor and gives a slightly blistered texture—finish in the oven during the last 15 minutes of the chicken's roast.

Make the mashed potatoes: Drain potatoes and return to the warm pot to dry for 1–2 minutes (this removes surface water and prevents watery mash). Heat the milk with 2 tbsp butter until warm (do not boil)—warming the dairy helps it absorb into potatoes for a creamier finish. Mash potatoes with a ricer or potato masher, then add warm milk and butter, salt to taste and 40 g (about 3 tbsp) Parmesan. Stir gently until silky; overworking will make them gluey because of released starches.

Finish the pan sauce (optional but recommended): When the chicken is done, transfer thighs to a cutting board and tent loosely with foil to rest 8–10 minutes (resting lets juices redistribute). Meanwhile, place the roasting pan on the stovetop over medium heat (use two burners if needed) and add 1 tbsp butter and the lemon-pan juices from the tray. Scrape browned bits (fond) and simmer briefly to deglaze, then swirl in the remaining 1 tbsp butter and taste—adjust salt, pepper and a drizzle of honey or a splash of extra lemon juice if you want brighter acidity. The quick pan sauce captures the best concentrated flavor from roasting and elevates the dish.

Assemble and serve: Spoon a generous mound of Parmesan-garlic mashed potatoes onto each plate or a family platter. Nestle a roasted chicken thigh on each mound, spoon some pan sauce over the chicken and scatter roasted green beans to the side. Garnish with a little extra chopped rosemary, a lemon wedge and a sprinkle of remaining Parmesan for a restaurant finish.

Serve family-style and enjoy: Encourage guests to squeeze extra lemon over their chicken for a bright lift. Leftovers keep well—chicken re-crisps in a 400°F oven for 10–12 minutes and mashed potatoes reheat with a splash of milk on the stovetop.