Before broiling, remove most of the tarragon and garlic from the chicken halves.
Arrange the chicken halves, end to end on the grate, directly under the broiler. Broil the chicken for about 15 minutes, basting with the excess marinade every 5 minutes. Re-arrange the halves after the first 10 minutes. Turn the halves over and broil for another 15-20 minutes, basting and re-arranging as before. Turn the chicken halves for the final time. Season with salt and pepper. Layer the cheese over both halves. Sprinkle with tarragon. Set the top grate up one level in the oven, and continue broiling until the cheese is bubbly. Remove from the oven, and allow to rest for 8 minutes.