Slice each chicken breast horizontally through the middle to create 4 thinner cutlets. If any pieces are thicker than 3/4 inch, pound with a meat mallet to even them out.
Set up two bowls. In the first bowl, crack the egg, add minced garlic, and a tiny pinch of salt and pepper. Whisk with a fork until combined. In the second bowl, spread the shredded parmesan in an even layer.
Pat chicken cutlets completely dry with paper towels. Dip each cutlet in the egg mixture, coating both sides, then let excess drip off.
Press the chicken cutlet into the parmesan, cut side down first. Press down hard to pack the cheese onto the meat. Flip and press the other side. Pack extra cheese onto any bare spots. Shake off loose bits. Repeat with remaining cutlets.
Heat a large non-stick skillet over high heat. Add butter and olive oil. Wait for butter to melt completely and foam to disappear.
Place coated cutlets in the hot pan. Press each cutlet down gently once with a spatula. Do not move them for 3 minutes.
After 3 minutes, check for deep golden brown color. Flip with tongs and cook the second side for 3 minutes, pressing down once. Total cooking time is 6 minutes.
Remove from pan and serve immediately with lemon wedges. The crust should be crispy and golden.