Preheat your oven to 170°F.
Dry the chicken and season with the salt and pepper. Put the chicken on a rack over a sheet pan in the refrigerator for 30 minutes to an hour then remove from the refrigerator and prepare your pan for cooking.
Melt 2 tablespoon of the butter in a large stainless-steel pan over medium-high heat, then add the olive oil. Add the chicken and brown well on both sides – about 6 minutes in total since these are fairly thin. The chicken should be about ¾ cooked. Remove the chicken to a plate and cover loosely with foil and set aside.
To the same pan, add the vinegar and garlic and deglaze the pan. Then add the chopped tomatoes and stir. Reduce heat to medium and cook, covered, for about 5 minutes. Remove the lid, add the chicken stock and tarragon stalks, and reduce the sauce by half over medium heat. This will take about 10 minutes.
Once reduced, remove the tarragon stalk and blend the sauce with an immersion blender or carefully pour into a regular blender and blend until very smooth. Return to the pan and continue reducing for another 3-5 minutes, until total volume has been reduced by ⅔.
Turn the heat to low and return the chicken and any juices on the plate to the pan, cover and simmer for 5 minutes to warm up and continue cooking the chicken. Remove the chicken again to a heat-proof dish, cover and keep warm in the oven.
Turn the heat off the sauce – making sure it is not boiling. Remove the lid from the sauce and whisk in the remaining 8 tablespoon of butter, 1 tablespoon at a time. Once incorporated, whisk in the crème fraîche or mascarpone.