Season the chicken breasts lightly with salt and pepper, then spread or coat each piece with the pesto sauce on all sides.
Place the flour on a plate and dredge both sides of each pesto-coated chicken breast in the flour, shaking off any excess.
Dip the floured chicken into the beaten egg mixed with 1 tablespoon water, ensuring it is fully coated.
Combine the Italian seasoned bread crumbs and shredded Parmesan cheese on a plate, then press each egg-coated chicken breast into the crumb mixture to fully coat.
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 breaded chicken to the skillet and cook 4 to 5 minutes per side, or until the chicken is golden and cooked through (internal temperature 165°F / 74°C).
Remove the chicken from the skillet and let rest a few minutes before serving.