To make the cutlets, slice each chicken breast in half lengthwise.
Place the chicken pieces between two sheets of plastic wrap and use the smooth side of a meat pounder (I like a rubber mallet) to gently pound them to about 1/4-inch thick cutlets.
Spray a not non-stick skillet with non-stick spray, and add 2 teaspoons of the oil. Heat the pan over medium-high heat until hot (about 2 minutes), then reduce the heat to medium.
Season each cutlet with coarse salt and fresh ground pepper. Dredge the cutlets in flour and add them to the hot pan. Cook the cutlets until they are golden brown and cooked through, about 2 1/2-3 minutes on each side.
Transfer them to a plate and cover the plate loosely with foil to keep them warm.
Add the remaining 1 teaspoon oil to the pan. Add the onion and saute until it's softened, about 5 minutes.
Deglaze the pan by adding the white wine and scraping all of the browned bits off the bottom of the pan (flavor, flavor, flavor).
Simmer until the wine is almost gone, about 5 minutes.
Add the chicken broth and lemon juice, bring it all to a simmer, and reduce it by almost half, about 5 minutes.
Add the tarragon and simmer for two more minutes.
Add the cream, cook 1 minute more;
stir in the peaches, and remove the pan from the heat.
Add the cutlets back to the pan and coat them with the sauce. Plate and enjoy!