Season chicken liberally with salt and pepper. Heat a large saute pan over medium-high heat, add oil, and place chicken in pan skin side down. Sear for 4 minutes, then reduce heat to medium and continue cooking for another 8 minutes. Turn chicken and cook another 4 minutes, or until just cooked all the way through.
Meanwhile, make sauce by melting butter in a medium saute pan over moderate heat. Cook shallots and mushrooms until soft and golden, about 10 minutes. Add jalapenos and cumin, lower heat, and cook for 5 minutes. Add chicken stock. Turn heat to high and cook until liquid is reduced by half. Add cream or half-and-half and cilantro stems and return to a boil. Remove from heat. Puree in a blender, pass through a wire mesh strainer, and return to heat.
Whisk egg yolks, sugar, and vinegar together in a small bowl. While continuing to whisk, pour one cup pureed cilantro sauce into egg mixture to temper. Then combine egg mixture with puree and cook over low heat, stirring constantly, until sauce is thick and smooth.
Arrange grilled chicken over a bed of your favorite rice and spoon sauce over all. Garnish with cilantro leaves, grilled scallions, and Pickled Tomato Salsa. Serve immediately.
Pickled Tomato Salsa:
In a large bowl, toss tomatoes with scallions and serrano chiles.
In a medium saucepan, bring vinegar to a boil. Add sugar and salt, and cook until dissolved, about 1 minute. Remove from heat and reserve. Measure ginger, garlic, mustard seeds, cracked peppercorns, cumin, cayenne, and turmeric onto a plate and place near stove. In another medium saucepan, heat oil over moderate heat until just smoking. Add spices and cook, stirring constantly with a wooden spoon, until aromas are released, about 2 minutes. Remove from heat and stir in vinegar mixture. Immediately pour over reserved tomato mixture. Stir to combine, cover with plastic wrap, and refrigerate 3 to 4 hours or several days. Makes 2 cups.