Melt 2 tablespoons of the butter in a large Dutch oven or deep heavy skillet over medium heat. Add the chicken pieces and sprinkle with kosher salt and pepper. Brown the chicken, turning to brown both sides. Remove the browned pieces to a plate and set aside.
Melt the remaining butter in the pan and add the sliced onions. Cook the onions for about 3 to 4 minutes, or until softened, stirring frequently. Add the garlic and cook, stirring, for 1 minute longer. Add the chicken pieces back to the pot and spoon some of the onion mixture over them.
Add the wine and chicken broth and tuck the rosemary around the chicken pieces. Bring to a simmer.
Cover, transfer to the oven, and bake for 25 minutes. Remove the cover and continue cooking for 10 to 15 minutes, or until the chicken registers at least 165 F on an instant-read thermometer inserted into the meat (not touching bone).