Place each chicken breast between two sheets of plastic wrap and pound to an even thickness of about 1/2 inch using a meat mallet or rolling pin.
Season both sides of the chicken breasts with salt and black pepper.
Set up three shallow bowls: one with the all-purpose flour, one with the eggs beaten together with water, and one with the panko breadcrumbs mixed with grated Parmesan cheese.
Dredge each chicken breast first in the flour, shaking off any excess, then dip into the egg mixture, and finally coat evenly with the panko-Parmesan mixture, pressing gently to adhere.
Heat olive oil and canola oil together in a large skillet over medium heat until shimmering.
Add the coated chicken breasts to the skillet and cook for 6-7 minutes on each side, or until the crust is golden brown and the chicken is cooked through (internal temperature should reach 165°F / 74°C).
Transfer the cooked chicken katsu to a paper towel-lined plate to drain any excess oil.
Slice the chicken into strips and serve immediately with tonkatsu sauce and lemon wedges.