Set up three shallow bowls: one with the flour, one with the beaten eggs, and one with the panko breadcrumbs.
Season both sides of each butterflied, pounded chicken breast with salt and pepper.
Dredge each chicken cutlet in flour, shaking off excess, then dip in the beaten eggs, and press into the panko to coat completely.
Heat 1/4 cup canola oil in each of two large skillets over moderately high heat until shimmering.
Add the coated chicken to the skillets and cook, turning once, until golden and crispy, about 3 minutes per side.
Transfer cooked chicken to a paper towel-lined baking sheet and sprinkle with a little salt.
Meanwhile, melt the 6 tablespoons butter in a small saucepan over moderately high heat and cook until it browns and smells nutty, about 4 minutes.
Stir the capers, lemon juice and chopped parsley into the browned butter.
Spoon the butter-caper-lemon sauce over the chicken cutlets and serve immediately.