Pat chicken breasts dry and season both sides lightly with salt and pepper.
Spread about 1/2 cup pesto sauce over both sides of each chicken breast so they are evenly coated.
Place the flour on a plate. In a shallow bowl, whisk the egg with 1 tablespoon water. On a third plate, combine the Italian-seasoned breadcrumbs and shredded Parmesan.
Dredge each pesto-coated chicken breast in flour, shaking off excess, then dip into the egg mixture, and finally press into the breadcrumb-Parmesan mixture to coat both sides.
Heat the olive oil and butter in a large skillet over medium heat until the butter is melted and the pan is hot.
Add the coated chicken to the skillet and cook 4 to 5 minutes per side, until the crust is golden and the chicken is cooked through (no pink remains and internal temperature reaches 165°F/74°C).
Transfer chicken to a plate and let rest a few minutes before serving.
