Preheat the oven to 350° F (18ºC). In a large, heavy-bottomed saucepan over medium-high heat, melt 1 tablespoon of the butter in 1 tablespoon of the olive oil. Season the chicken all over with salt and pepper. Working in batches as needed to avoid crowding, add the chicken to the hot pan, skin side down, and cook, turning once, until golden brown on both sides, about 6 minutes per side. Transfer each batch, skin side up, to a 9-by-13-inch (23-by-33-cm) baking dish as it's finished. Set aside. Do not wipe the pan clean and keep it over medium-high heat. 

Add the garlic to the saucepan and cook, stirring often, until lightly browned and beginning to soften, 2-3 minutes. Using a slotted spoon, spread the garlic all over the chicken. 

Add the wine to the hot saucepan, scraping up any browned bits on the bottom of the pan, and bring to a simmer. Cook until the liquid reduces by half, about 2 minutes. Add the broth and return to a simmer. Add the remaining 1 tablespoon butter. When the butter has melted, season well with salt and pepper and stir to combine. Carefully pour the sauce all over the chicken and garlic. 

In a bowl, toss the potatoes with the remaining 1 tablespoon olive oil and season with salt and pepper. Tuck the potatoes in the baking dish all around the chicken. Scatter the thyme sprigs over the top. 

Roast until the potatoes are fork-tender and the chicken is opaque throughout (no pink should show when you cut in next to a bone, and the chicken juices should run clear), 35-40 minutes. Serve right away.