How to Make It
Melt the butter in a medium skillet. Add the mushrooms and shallots and cook over moderate heat until soft and lightly golden, about 8 minutes. Add the chopped cilantro, jalapeños and cumin, reduce the heat to moderately low and cook for 5 minutes. Add the chicken stock, raise the heat to high and simmer until the liquid is reduced by half. Add the cream and bring to a boil. Remove from the heat and, working in batches, puree the sauce in a blender or food processor. Strain the sauce and return it to the skillet.
Light a grill or preheat the broiler. Season the chicken lightly with salt and pepper. Mix the oil and lime juice in a small bowl and brush all over the chicken and scallions. Grill or broil the chicken, skin side down, until browned and cooked though , about 6 minutes per side for thighs and 5 minutes per side for breasts. During the last 2 minutes, grill or broil the scallions until tender and lightly charred. Transfer the chicken and scallions to a platter; keep warm.
Reheat the sauce. In a bowl, mix the egg yolks, sugar and vinegar. Gradually whisk 1 cup of the hot sauce into the egg mixture. Pour the egg mixture back into the skillet and stir over low heat until thickened and smooth, 2 to 3 minutes; don't let the sauce boil. Spoon the sauce over the chicken and garnish with the scallions and cilantro leaves. Serve with the Spicy Tomato Relish and yogurt.