In a shallow bowl stir together flour, salt and pepper. Set aside. Cut off fillet from chicken breast if attached. Slice each chicken breast horizontally into two pieces. Cut each piece in half, giving 4 pieces of similar size per chicken breast, plus the fillet if included. Place each piece, including fillet between two pieces of parchment paper. Pound with a flat mallet until 1/4" thick.
Heat 1 tbsp oil in a large skillet over medium high heat. Toss chicken pieces in flour and shake off excess. Fry half of the chicken pieces 2 minutes per side and remove to plate and keep warm. Repeat with remaining oil and chicken.
Reduce heat to medium and add 1 tbsp of butter to the pan, add chopped shallot and sauté for 1 minute until softening but not browning. Add wine, stirring and scraping the pan to remove all the cooked bits. Stir in broth and reduce for 3-4 minutes until about 1/2. Whisk in mustard and remaining butter, small chunks at a time, allowing each to incorporate before adding more. Stir in lemon juice and tarragon. Season with salt and pepper. Add chicken cutlets back to pan and spoon with sauce. Serve immediately.