Butterfly the chicken (slice each piece in half widthwise, so they are half as thick as they were to begin with). If the pieces are still uneven (thicker on one part than another), lay them on a cutting board, top with parchment paper, and use a meat mallet or rolling pin to pound them to an even thickness. Season the chicken breast on both sides with salt and pepper.
Place the flour, eggs, and panko in three separate large, shallow bowls. Dredge the chicken breasts in flour on both sides, shake off the excess, then dip in the eggs, and then the breadcrumbs. Place on a tray or cutting board while you finish the remaining chicken.
Heat the olive oil in a large, sturdy skillet over medium high heat. Add as many of the dredged chicken breasts as you can fit without overlapping, and fry for 2-3 minutes a side or until golden brown and cooked through. Remove to a plate. Repeat with remaining oil and chicken breast, adding additional oil if needed.
When the chicken has been removed, add the garlic to the skillet and cook for just a minute. Add the broth, scraping up any browned bits at the bottom as you go. Allow the broth to simmer until reduced by half, about 5 minutes.
Lower the heat to medium low and add the butter, stirring until melted. Stir in the capers, lemon juice, and lemon slices, if desired.
To serve, plate the chicken then spoon the lemon butter sauce over top. Top with lemon slices and fresh parsley.