Dice the onion. Quarter the mushrooms. Blot away the excess moisture on the hen with paper towels. Heat up a Dutch oven over a medium flame. In 2 Tbl of clarified butter, brown the hen - on all sides. Remove the bird and degrease the pan. Add 2 Tbl of clarified butter. Stir in the flour and whisk it until its rawness dissipates - about 2 minutes. Adjust the flame to medium-low. Sweat the onions down in the roux. When the onions are translucent, throw in the white wine, cayenne and bay leaf. Season to taste with salt and pepper. Return the hen to the pot. Bring the contents of the pan to a boil, cover and simmer slowly for 25 minutes. Add the mushrooms. Cover and simmer for about 15 minutes.
Remove the hen and bay leaf from the pot. Increase the heat. Reduce the sauce (if necessary). Pour in the heavy cream. Reduce until the sauce barely coats the back of a spoon. Test for seasoning. Serve the hen and sauce over rice.