Dredge the pounded chicken breasts in flour.

Brown Chicken:

In a large skillet, melt 2 tablespoons of butter over medium-high heat.

Brown the chicken breasts in the butter until golden brown on both sides.

Once browned, remove the chicken from the skillet and transfer to a warm platter.

In the same skillet, add the remaining 2 tablespoons of butter.

Add the sliced mushrooms and sauté until they are golden brown and tender.

Sprinkle the mushrooms with 2 tablespoons of flour, stirring to blend well.

Gradually pour in the chicken broth, stirring constantly.

Add the minced fresh tarragon or dried tarragon.

Cook the mixture until it thickens.

Slowly pour in the half and half, stirring continuously.

Season the sauce with salt and freshly ground black pepper to taste.

Return the browned chicken breasts to the skillet, nestling them into the sauce.

Heat the chicken through in the sauce.

Once heated through, serve the tarragon chicken hot, spooning the sauce and mushrooms over the top.