Put oven rack in middle position and preheat oven to 450°F Pat hens dry and sprinkle with salt and pepper.
Roast, cut sides down, in a large heavy shallow baking pan (1 inch deep) until just cooked through, about 30 minutes.
While hens roast, simmer vermouth, garlic, thyme, and 1/2 cup water, uncovered, in a 2-quart heavy saucepan until garlic is soft, about 15 minutes. Discard thyme.
Mash garlic into sauce with a fork and simmer until reduced to a glaze (about 3 tablespoons).
Brush glaze onto hens and roast 5 minutes more. Pour remaining 1/2 cup water into saucepan, swirling to dissolve any remaining glaze, and reserve.
Transfer hens to a serving dish and let stand, loosely covered with foil, 5 minutes.
Straddle baking pan over 2 burners, then add reserved liquid from saucepan and deglaze baking pan by boiling over moderate heat, scraping up brown bits, until reduced slightly. Season pan juices with salt and pepper and pour over hens.