Set out three shallow bowls: one with the flour, one with the beaten eggs, and one with the panko breadcrumbs.
Season the butterflied, pounded chicken cutlets on both sides with salt and freshly ground black pepper.
Dredge each cutlet in flour, shaking off excess, then dip into the beaten eggs, and press into the panko to coat evenly.
In each of two large skillets, heat 1/4 cup of canola oil over moderately high heat until shimmering.
Add the coated cutlets (work in batches if needed) and cook over moderately high heat until golden and crispy, about 3 minutes per side; turn once.
Transfer cooked cutlets to a paper towel-lined baking sheet and sprinkle lightly with salt.
Meanwhile, melt the butter in a small saucepan over moderately high heat and cook until it browns and smells nutty, about 4 minutes, swirling occasionally.
Remove the saucepan from heat, stir in the capers, lemon juice, and chopped parsley.
Spoon the brown butter, caper, and lemon sauce over the schnitzels and serve immediately.
