In a shallow dish, combine flour with a pinch of salt. In another dish, whisk together egg, 1 tablespoons water, and a pinch of salt. In a third dish, stir together breadcrumbs, 1/2 teaspoons salt, and 1/4 teaspoons black pepper.
Working with one cutlet at a time, dredge cutlet in flour to coat completely, then shake off excess flour. Dip floured chicken in the egg mixture, turning to coat completely. Let excess drip off. Place cutlet in breadcrumb mixture, patting it in and turning it coat both sides. Shake off any excess crumbs before proceeding.
In a large nonstick skillet, heat 2 tablespoons oil over medium; swirl to coat pan. Add 4 cutlets; cook until golden brown and no longer pink in the center, 2-3 minutes per side. Remove. Repeat with remaining oil and chicken. Remove cutlets to serving tray. Squeeze juice of two lemons into bottom of skillet, add 2 tablespoons butter and swirl until melted, scraping up any browned bits. Serve sauce with cutlets.