Prep the chicken: Working with 1 ½ pounds chicken breasts halved lengthwise, place each cutlet, one at a time, between two sheets of plastic wrap and use a meat mallet or rolling pin to pound out to ½-inch thickness. In a shallow bowl, mix 2 tablespoons flour, ½ teaspoon kosher salt, and ½ teaspoon black pepper. Lightly dredge both sides of each chicken cutlet in the flour, shaking off the excess, and set aside.
Brown the chicken: Heat a large cast iron pan or heavy bottom skillet over medium and melt 2 tablespoons butter with 1 tablespoon canola oil. Place chicken cutlets into the pan and cook, undisturbed, until browned on underside, about 4 minutes. Turn the chicken and cook for 3-4 minutes on the second side. Transfer to a plate, tent with foil, and repeat with the remaining chicken cutlets, adding more butter or oil as needed.
Make the sauce: Wipe out any burned bits from the skillet with a paper towel and return to medium-high. Add ½ cup chicken broth or wine to the pan and bring to a boil. Scrape up browned bits of flavor, cook for 2 minutes, and add 1 cup cream. In a small bowl, mash remaining 1 tablespoon flour and 1 tablespoon butter until smooth, then whisk into the chicken broth. Reduce the heat and simmer until thickened, 3-5 minutes or until the cream coats the back of a spoon. Stir in juice and zest of 1 lemon, 2 tablespoons chopped fresh tarragon, and adjust seasoning if needed.
Re-warm the chicken: Return chicken breasts to the pan and simmer for 2 minutes or until chicken is warmed through. Serve with rice, noodles, or mashed potatoes with more chopped tarragon or lemon wedges on the side.