In a large sauté pan over medium high heat, add the butter and sauté until the foam subsides and the butter begins to brown.
Add the onions, the mushrooms, the salt, and a few cranks of pepper, then toss to coat.
Cook, stirring occasionally, until the onions are transparent, for 7 minutes.
Turn the heat to high, add the sliced chicken, and stir to combine.
Dust the mixture evenly with the flour, toss to coat, then cook, stirring occasionally, for 2 additional minutes.
Slowly drizzle the milk over the chicken mixture, stir to combine, and bring to a simmer.
Lower the heat, add the cheese, the tarragon, and the parsley and stir to combine.
Cook until the sauce thickens, about 2 minutes, and season to taste.
Serve with brown rice, egg noodles, or steamed vegetables, if you wish.