Heat your oven to 300 degrees.
I like to pound out chicken breasts just so that they are an even thickness. This is a personal preference, you can skip this step.
Mix the pesto and eggs together in a shallow bowl, being sure to completely break up the eggs.
Pour the Parmesan and panko into another shallow bowl. Stir to combine.
One by one, take the chicken breasts and coat them in the Parmesan and panko mixture, lightly pressing it into the chicken breast. Then carefully dunk it in the pesto and egg mixture. Place the chicken breast back in the Parmesan and panko mixture, fully coating it. Continue this with all of the chicken breasts.
Heat a skillet over medium heat, and add enough butter to coat the bottom of the skillet. Add 2 chicken breasts - it is likely you can only fit two chicken breasts in your skillet at a time. Let the first side of the chicken cook for 3-4 minutes, until golden brown, then flip to brown the second side. Once both sides of the chicken have browned, place the chicken on a baking sheet.
Continue this same process until all of the chicken breasts have been browned.
Bake the chicken for 20-25 minutes, or until it is no longer pink in the center. Serve right away.