Preheat oven to 350 degrees.
Trim duck breasts if necessary. Place in pan. In a separate bowl, mix 1/2 of the red wine vinegar, olive oil, bay leaf, thyme and shallots. Season with salt and pepper. Pour over duck and marinate for 30 minutes.
Trim top portion of the stem on figs. Season with pepper. Place in pan and add remaining red wine vinegar and red wine. Hold aside.
Heat grill. Grill duck until cooked to desired doneness. Brush with marinade while grilling. Reserve leftover marinade and heat on stove until it just comes to a boil. Hold aside.
Take duck off grill and let rest. Place figs on grill to warm, skin side down. Slice duck on the bias. Divide arugula into 5 portions and put on plate. Place duck and figs on top. Sprinkle with blanched chestnuts. Drizzle cooked marinade over duck, figs and greens.