Preheat oven to 375 degrees.
Add peaches and cherry tomatoes to a medium bowl. Sprinkle with 1/2 teaspoon salt and 1/8 teaspoon pepper. Stir to coat. Give them a stir every 5-7 minutes while the chicken is cooking.
Add chicken to a plastic ziplock. Use a meat mallet or rolling pin to flatten the chicken to 1/2-inch thickness. The thicker the breast the longer it will take to cook through.
Lay the chicken down horizontally with the tip of the breast pointing to your right. Use a pairing knife to gently cut a slit in the top of the breast, this will be to your left. Use your knife to make a pocket in the breast but do not cut through to edges. Stuff the cubed mozzarella in the pocket and lay as flat as possibly.
Add the flour to a medium plate or shallow bowl. Season with salt and pepper. Add the egg to another shallow bowl along with two tablespoons of water. Whisk to combine. Add the panko, parmesan, granulated garlic, and remaining teaspoon salt to another shallow bowl or plate. Mix to combine.
Season the chicken breasts on both sides with a little bit of salt and pepper. Dredge in the flour. Pat off excess flour. Dip in the egg. Let any excess drip off. Coat in the panko. Pat off any excess panko.
Heat a large oven-proof skillet to a medium heat. Add three tablespoons butter and olive oil. Once the butter melts and starts to foam, add the chicken. Pan-fry for 4-5 minutes until golden brown and crisp. Flip and do the same thing on the other side. Use an instant read thermometer to check the temperature of the chicken. It's done when it reaches 165 degrees. If both sides of your chicken are already brown and crisp and it has not reached 165 degrees, pop in the oven until it does. The time this takes depends on the size of your chicken. For one batch I had thicker breasts and I had to bake them for an additional 5-6 minutes. For the other, I didn't need the oven.
Remove the chicken from the pan, set aside. I like to store them on a baking sheet lined with a cookie cooling rack so the bottom doesn't get soggy. Set the empty pan over a medium-high heat. Add remaining butter to the pan. When the butter starts to brown, swirl the pan. Swirl the pan until the butter turns a deep golden brown and smells nutty. Remove from the heat and add lemon juice and thyme leaves. Stand back, the mixture will spit and spatter. It will calm down in a few seconds. Stir to combine. Season with salt and pepper.
Spoon the peach and tomato mixture over the chicken. Spoon the brown butter sauce all around. Garnish with fresh thyme leaves and flaked Maldon salt.