Melt 1 tbsp butter in a medium non-reactive saucepan. Saute onions until translucent, 2-3 minute Add wine, juice and broth. Turn heat to high until sauce boils, then turn down to medium and reduce to 1 1/2 cups. Strain into a measuring cup and set aside.
Preheat oven to 200 degrees F. Place a baking rack in a rimmed baking sheet and place in oven.
Slice breasts in half horizontally and pound to an even 1/4" thickness. Whisk salt and pepper into flour in a pie plate. Whisk eggs and milk in a second pie plate. Dip chicken cutlets in flour, then in egg, then back in flour. Set aside on a rack over a baking sheet.
When all 8 cutlets are coated, melt 1 tbsp butter in 1 tbsp oil in a 12" nonstick skillet. Saute 4 cutlets over medium heat until nicely browned, 1 1/2 to 2 minutes. Flip and brown another 30-60 seconds. Remove cutlets to baking rack in oven.
Wipe out pan with paper towel, and using another 1 tbsp butter and 1 tbsp oil, cook remaining 4 cutlets. Remove to baking rack.
Wipe out pan with paper towel and add reserved sauce to now-empty pan over medium heat. Swirl in remaining 2 tbsp butter. Dip cutlets in sauce and serve, 2 cutlets per serving.