Trim any fat from the chicken breasts and slice them horizontally in half.
Grate the zest from one of the lemons and juice both lemons. You need about ¼cup juice. Finely chop enough of the parsley leaves to measure 1tablespoon.
Put 2tablespoons of the butter and the vegetable oil in a large skillet and place over medium-high heat.
Spread the flour on a small plate. Coat the chicken fillets with flour, shaking off the excess. When the butter in the pan is no longer foamy and the chicken sizzles when a corner is placed in the pan, carefully slide the chicken into the pan. Put in only as many fillets as will comfortably fit in the pan at once. When one side is lightly browned, turn the fillets and brown the second side. The chicken should take 3 to 4 minutes to cook, depending on the thickness of the fillet. When done, the chicken will feel firm when prodded with tongs. Transfer to a platter and continue until all the chicken is cooked. Season the fillets with salt and pepper.
Lower the heat to medium-low and add the lemon juice and zest. Season lightly with salt and cook until the sauce is thick enough to coat a spoon, 1 to 2 minutes. Swirl in the remaining 1tablespoon butter and add the parsley. Return the chicken to the pan and turn the fillets in the sauce. Serve at once.