Preheat oven to 425 degrees.
Lay the tomatillos (cut side down) in a very large non-stick skillet along with the garlic and serranos. Place the skillet over medium-high heat. When the tomatillos are well browned, turn them over and brown the other side. The tomatillos should be completely soft. Since the garlic and serranos brown more quickly, you'll need to turn them individually and remove them, if necessary. Scoop everything into a blender jar or food processor, along with 2/3 cup cilantro and 1 teaspoon salt. Blend until smooth. Pour the sauce into the bottom of a 13 x 9 casserole dish.
While the tomatillos are roasting, toss the chayote and the onion with 1 tablespoon oil and 1/2 teaspoon salt and spread onto a baking sheet. Place in the preheated oven and roast for 30 to 35 minutes or until the chayote is beginning to turn brown around the edges. You should stir the mixture several times during the roasting process. Scrape back into the bowl and mix in the sliced shiitakes.
Sprinkle the chayote mixture over the top of the sauce in the casserole dish. Lay chicken on top, skin-side up. Set baking pan in middle of oven. Bake 30 to 35 minutes, until chicken is lightly browned and cooked through. To tell if chicken is done, poke a fork into the chicken near the bone -- if the juice that runs out is clear, the chicken is done—if pink, bake longer.
Use tongs to move chicken to serving platter. Stir in the cream and sugar into sauce. Taste and add more salt and sugar if you think it needs it. If the sauce is too thick, stir in a little water to thin it out. Spoon sauce over chicken, decorate with cilantro sprigs and serve.