Cut chicken breast horizontally through the middle to within one-half inch of the other side. Open the two sides and spread them out like an open book; cut into 4 pieces. Cover chicken pieces with plastic wrap; pound firmly with a meat mallet to an even thickness.
Dust chicken pieces lightly with flour on both sides.
Whisk eggs in a shallow bowl until smooth. Dip floured chicken in eggs until coated.
Heat oil in a deep skillet over medium-high heat. Cook chicken in the hot oil until golden, 2 to 3 minutes per side. Drain on paper towels.
Combine butter, parsley, salt, and pepper in a skillet; heat over medium heat until butter is melted, 1 to 2 minutes. Add chicken pieces; swirl skillet to coat them with butter. Pour in lemon juice; swirl skillet to coat chicken evenly.
Pour white wine into the skillet. Cook until reduced, about 2 minutes. Pour in chicken stock. Cook until chicken is no longer pink in the center, about 10 minutes.