Preheat oven to 350 degrees.
Place the chicken between plastic sheets or preferably Silpats and pound with a heavy skillet to 1/4 inch thickness. I like to do this with a large value pack and freeze the extras, individually, after sealing with the FoodSaver . This way I have this done for the next few times I want this recipe or a Chicken Cordon Blue. Place the chicken breasts on a sheet pan laid out flat. Have toothpicks ready on the pan to hold them shut.
On a separate plate, lay out the 4 pieces of prosciutto, basil leaves and with a small ice cream scoop dish out 4-1 oz scoops of goat cheese. First place the prosciutto on each breast, then put the goat cheese on the prosciutto (spread out some with the scooper) then put the basil leaves on the cheese and press down to spread out evenly. Roll the chicken breasts up and put a tooth pick in to hold.
In a heavy sauce pan, melt 2 tbs butter. Add the garlic and cook until soft. Add the white wine and 1-1/4 cup chicken broth. Bring to a boil and reduce until half.
While the sauce is reducing, heat on medium high 1 tbs olive oil and 1 tbs butter in an oven proof skillet. Add the chicken rolls. Braise the chicken on all sides, then add 1/2 cup chicken broth to the pan and place in the oven. The extra broth helps to keep the chicken moist while it is cooking through in the oven.
Once the chicken is cooked though, remove from the oven and place on a plate and cover loosely with aluminum foil. Pour the contents of the skillet into the sauce and sauce and stir. Add the creme fraiche and the the tarragon and lemon and stir.
Place a small ladle full of sauce on a plate and put the chicken roll in it.
Serve with a rice dish (rice pilaf) and steamed vegetables.