Melt the butter in a large skillet over medium heat. Stir the tarragon, nutmeg, green onions, and 2 pinches of salt into the melted butter; cook and stir until the green onions are softened, about 4 minutes. Add the mushrooms, stir to coat, and continue cooking until the mushrooms are tender, about 5 minutes more. Fold the chicken into the mixture and cook together until the chicken gets hot, about 1 minute. Sprinkle the flour over the mixture; stir until completely absorbed into the mixture, 3 to 4 minutes.
Dissolve the beef bouillon cubes in the hot water; pour into the chicken mixture about 1/2 cup at a time, stirring to dissolve any clumps of flour between batches. Increase heat to bring the mixture to a boil; return heat to medium. Stir the sour cream into the mixture until well integrated.
Dissolve the cornstarch into the cold milk; stir into the soup. Increase the heat to medium-high and cook the soup until it begins to bubble on the sides, but not boiling; again return heat to medium, stirring continually. Add the lemon juice, cayenne pepper sauce, salt, and pepper; stir. Serve hot.
