In a medium saucepan, melt 2 tablespoons butter over medium heat. Add the flour and cook, stirring constantly, until thickened into a pale blond roux, about 2 minutes. Add the hot chicken broth in a steady stream, whisking to incorporate. Bring to a simmer and cook, stirring frequently with a whisk or heavy wooden spoon until thick. Add salt and pepper, and stir to incorporate. Slowly whisk in the milk and simmer, stirring, until well incorporated and thickened. Add the Gruyere cheese; stir until melted. Remove from the heat and very lightly film the top with the milk to prevent a skin from forming.
In a large skillet, heat the remaining 2 Tablespoons of butter over medium-high heat. Add the diced chicken, season lightly with Creole seasoning, and cook, stirring, until lightly browned and cooked through, about 4 minutes. Remove with a slotted spoon. Add the shallots and garlic to the fat and juices in the pan, and cook, stirring, for 1 minute. Add the mushrooms, and cook, stirring, until they have given off their liquid and are starting to color, 3 to 4 minutes. Add the ham and cook, stirring, for 1 minute. Add the sherry and bring to a boil, stirring to deglaze the pan. Add the tarragon and return the chicken to the pan. Stir well to blend and cook until the sherry has evaporated. Reduce the heat to low and stir in enough of the creamy white sauce to bind and thicken, about 3/4 cup. Remove from the heat and let cool slightly.