Pre-heat your oven to 170 degrees and place an oven proof plate in the oven to get warm.
Place a large heavy bottom pan on your stove and add your olive oil and butter and preheat over medium heat.
Now we are going to pound the chicken breasts thin.
Line a counter with a sheet of plastic wrap large enough to place all 4 chicken breast on. Place your chicken breasts on the plastic wrap and cover with a second piece of plastic wrap making sure all of the chicken is covered well. Take a rolling pin (or something similaand begin to pound down your chicken breasts until they are nice and thin and almost double in size.
Sprinkle ½ of your salt and pepper on the side of the chicken breasts facing up.
Now place your chicken breasts salt and pepper side down in your pan and cook for about 4-5 minutes a side until they are nice and browned.
While the chicken is cooking, sprinkle the other ½ of your salt and pepper on the chicken.
Flip your chicken breasts over and cook for another 4-5 minutes.
Once the chicken is done cooking add them to the warm plate in your oven and cover it with aluminum foil.
In the same pan that you cooked your chicken, over medium heat, add your diced shallots and a pinch of salt and cook until the shallots become translucent (about 4 minutes).
Now squeeze in the juice from 1 lemon and scrape up the little yummy bits from the pan. Your sauce will take on a light brown color at this point.
Now add your cream and fresh tarragon and mix well and turn heat to low.
Add your chicken from the oven back into the pan with the cream and cover.
Let this sit for about 4 minutes and remove from the heat.
Plate your chicken by adding it to a plate and spooning sauce over top of the chicken.