Do ahead (optional): Marinate chicken breasts in 2 tablespoon dijon mustard.

Melt butter and olive oil in skillet over medium-high heat.  Place chicken in skillet and brown on both sides.  Reduce the heat to medium, cover skillet, and continue cooking for 15 minutes.  

While the chicken is cooking, you can chop the tarragon, mince the garlic and shallots, and prepare any side you plan on serving with the chicken.  

After the chicken is done cooking, remove from skillet and set it aside while you make the sauce.  

Using the same skillet, add the garlic and shallots and saute for a couple of minutes, until the shallots start to turn translucent and the garlic becomes fragrant.  

Add the mushrooms and cook until softened, about 5-7 minutes. You might want to add a little bit more olive oil if the pan is pretty dry after you remove the chicken.  

Add the cream and scrape up any bits on the bottom of the pan.  Add the dijon and tarragon.  Simmer for 5 minutes, when the sauce starts to thicken.

Return chicken to pan and coat with sauce.  

Serve and enjoy!