If the chicken breasts are really thick pound the chicken breasts with a meat mallet to half an inch thick. Season on both sides with salt and pepper.
In a large skillet over medium heat, melt the butter. Add the chicken breasts and cook 4-5 minutes on each side until browned.
Remove the chicken from the pan and place on a plate. Keep warm in the oven.
Add the mushrooms and the minced shallot to the pan and saute until soft. If needed, add another tablespoon or two of butter at this point to make sure you have enough to fry the ingredients in.
Add in the minced garlic and saute until browned and fragrant, around 2 minutes. Add in the flour, mixing and frying in the mixture.
Deglaze the pan with the white wine, slowly pouring the wine into the pan and scraping the browned bits off of the bottom of the pan with a wooden spoon. Go slow and make sure the flour doesn't get lumpy.
Add in the chicken stock then cream and stir until combined, and add in the fresh tarragon.
Place the chicken breasts into the pan in the sauce and spoon some sauce over them. Place a lid on top of the pan and cook on medium heat until the sauce is slightly thickened and the chicken is done when an instant read thermometer reaches 165 degrees.
Stir in the lemon juice and season with salt and pepper to taste.