Heat the olive oil in a large skillet, over medium heat for about 2 minutes or until almost smoking. Add the game hens, skin side down, and saute for 5 minutes or until golden. With tongs, turn the hens over, add the garlic and saute for another 5 minutes.
Add the vinegar and broth which will bubble up, then the thyme and rosemary. Cover and simmer, over low heat, for 25 to 30 minutes or until the juices run clear, not rosy, when the thigh is pricked with a fork. Transfer the hens to a plate. Stir the mustard into the juices in the skillet and boil down furiously, until 1/2 cup of liquid remains, scraping the brown cooking particles into the sauce. Season with salt and pepper, spoon over the hens and serve immediately.