Prep the chicken. Working with one chicken half at a time, place the chicken between two sheets of plastic wrap and use a meat mallet or rolling pin to pound out to ½-inch thickness. In a bowl, mix flour, salt, and pepper then lightly dredge chicken in flour, shaking off the excess.
Brown the chicken. Heat a large cast iron pan or heavy bottom skillet over medium and melt 2 tablespoons butter with canola oil. Place chicken cutlets into pan and cook, undisturbed, until browned on underside, about 5 minutes. Flip the chicken and cook for 3-4 minutes on second side. Transfer to a plate, tent with foil, and repeat with the remaining chicken cutlets, adding more butter or oil as needed.
Cook the mushroom and onion. In the same skillet, wipe out any burned bits then melt 2 tablespoons butter and cook mushrooms and onions until soft, 2 to 3 minutes. Add lemon juice, transfer to a bowl, and set aside.
Make the creamy sauce and bring it all together. Return pan to medium-high heat, add chicken broth, bring to a boil, and scrape up any browned bits. In a small bowl, mash flour and 1 tablespoon butter until smooth, then whisk into the chicken broth. Reduce heat and simmer until thickened, about 5 minutes. Stir in cream, reserved mushroom mixture, and chives. Nestle cooked chicken and drippings into the sauce. Serve over rice, noodles, or mashed potatoes.