To make the chimichurri sauce, pulse the garlic in a food processor fitted with the metal blade until it is minced. Add the parsley, vinegar, oregano, red pepper flakes, paprika, salt, and pepper. Process until finely chopped, about 1 minute, scraping down the sides of the bowl as needed. With the processor running, add the olive oil in a slow, steady stream until completely combined. Scrape down the sides of the bowl and pulse 2 to 3 times more. Transfer the sauce to an airtight container and refrigerate until ready to use, or up to 1 day.
Place each chicken breast between two pieces of plastic wrap and gently pound it with a meat mallet until it is evenly 1/4 inch thick. Cut each breast in half (as you cut the breast in half, the pounded breast will open up like a book) to make 4 cutlets. Sprinkle each cutlet with the salt and pepper.
In a shallow dish, whisk together the eggs, garlic, 2 tablespoons of parsley, and water.
Place the bread crumbs in another shallow dish.
In a large heavy skillet, heat 1/4 inch of pure olive oil over medium heat to 315˚F. Set a wire rack over a rimmed baking sheet.
Working with 2 at a time, dip the cutlets in the egg mixture, press them into the bread crumbs to coat evenly on both sides, and then carefully place them in the hot oil. Fry for 4 to 5 minutes on each side, or until cooked through and juices run clear. Drain the chicken on the wire rack. Repeat with the remaining chicken. Serve with the sauce.