Trim each chicken thigh: remove excess fat and loose skin from the back while leaving skin covering the top. Pat the thighs dry with paper towels.
Trim both ends from the green beans. Put the trimmed beans in a bowl of ice water to crisp them while you prepare the chicken.
Position a rack in the center of the oven and preheat to 425°F (220°C).
Drain the green beans and spin or pat them dry (salad spinner or clean kitchen towel). Return the beans to a bowl and set aside.
Arrange the chicken thighs skin-side up on a large rimmed baking sheet in a single layer, leaving space between pieces. Let the chicken sit at room temperature for about 15–30 minutes while the oven finishes preheating.
Roast the chicken on the center rack for 20 minutes.
Remove the baking sheet from the oven. Push the chicken pieces toward the edges of the sheet to create space in the center, then spread the dried green beans in a single layer in the open space between and around the thighs.
Return the sheet to the oven and continue roasting for 20 more minutes, or until an instant-read thermometer inserted into the thickest part of a thigh (not touching bone) registers at least 165°F (75°C). If thighs are very large, continue roasting and check temperature again as needed.
Remove the pan from the oven and let the chicken and green beans rest for 5 minutes before serving hot.