Slice the chicken breast in half lengthwise to create thinner cutlets. Using the bumpy side of a meat mallet, pound the chicken until ¼-inch (6 mm) thick.
Add the flour, eggs, and bread crumbs to 3 separate, shallow dishes.
To the dish with the eggs, add the parsley, garlic, and milk. Season with salt and pepper. Mix with a fork to combine.
Add the Parmesan cheese to the bread crumbs and stir to combine.
Working 1 piece at a time, coat the chicken cutlets in the flour, shaking off any excess, then the egg mixture, then the bread crumbs. Set the coated chicken on a baking sheet lined with parchment paper and refrigerate for 1 hour.
Melt the butter in a large saucepan over medium heat. Add enough olive oil to come ¼ inch (6 mm) up the sides of the pan. Working in batches, fry the chicken until golden brown, about 3-4 minutes per side. Add more oil as needed for subsequent batches. Drain the chicken on paper towels to absorb any excess oil. Season with a pinch of salt.
Serve the chicken with lemon wedges. The dish pairs well with pan-roasted potatoes and tomato salad.