1Cut or tear the stale sourdough into chunks. Pulse in a food processor until you get fine to medium crumbs (about 3–4 cups). Transfer to a bowl or shallow dish. Stir in the dried oregano and black pepper until evenly distributed.
2Pat the chicken breasts dry with paper towels. Butterfly each breast by slicing horizontally through the thickest part, opening it like a book and cutting all the way through.
3Place each breast between two sheets of plastic wrap or parchment paper. Using the flat side of a meat mallet, pound gently but firmly until the cutlets are very thin (about ¼-inch thick or thinner, where light shines through). Season lightly with salt and pepper if desired.
4Prepare three shallow dishes or plates: one for all-purpose flour, one for beaten eggs, and one for the seasoned homemade breadcrumbs.
5Working with one cutlet at a time, dredge in flour, shaking off excess. Dip in beaten egg, letting excess drip off. Press into the breadcrumbs, coating both sides thoroughly and pressing gently. Place breaded cutlets on a plate or wire rack.
6Heat a generous layer of olive oil (about ¼–½ inch deep) in a large skillet over medium-high heat until shimmering (around 350°F).
7Fry the cutlets in batches (don’t overcrowd) for 2–4 minutes per side, until golden brown and crispy, and the internal temperature reaches 165°F.
8Transfer to a paper towel-lined plate or wire rack. Immediately sprinkle with salt while hot.