Rinse quail and pat dry.
Melt 2 tablespoons butter in a 10- to 12-inch frying pan over medium-high heat.
Add birds and brown lightly on all sides, about 10 minutes.
Breast meat should be red and moist at the bone; to check, cut from just above wing joint to breastbone.
Lift birds into a serving dish; keep warm.
To pan, add broth, vinegar, and rosemary.
Over high heat, boil, uncovered, until reduced to 1/2 cup, about 5 minutes.
Add remaining butter; stir constantly over medium heat until smoothly blended.
Pour sauce over birds and garnish with rosemary sprigs.