Melt the butter in a large pot over medium heat. Stir in the onions, and cook for a few minutes until they begin to soften. Add the chicken breast and thigh meat; cook and stir until the chicken is no longer pink in the center, about 10 minutes. Pour in the water and cream of chicken soup; season with chicken bouillon and poultry seasoning. Bring to a simmer over medium-high heat.

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Meanwhile, pour the milk into a mixing bowl, and stir in the baking mix until a dough forms. Once the soup is simmering, drop the dumpling mix in by the spoonful. Reduce heat to medium-low, cover, and simmer until the dumplings are no longer doughy in the center, about 20 minutes. Turn the dumplings over after 10 minutes to ensure they cook evenly.