Pat the chicken breasts dry and season them generously with salt and pepper. Heat the cast iron skillet over medium-high heat, then add olive oil. Place the chicken breasts in the skillet and sear until golden brown on both sides, about 4-5 minutes per side, until cooked through. Remove the chicken from the skillet and set aside.
Reduce the heat to medium and add more olive oil if needed. Toss in the minced garlic and grated ginger, cooking until fragrant—about 30 seconds—watching for bubbling and golden aroma.
Add sliced peaches to the skillet, arranging them in a single layer. Cook for about 3-4 minutes until they begin to soften and develop a light caramel color, releasing sweet fragrance into the kitchen.
Stir in honey and lemon juice, coating the peaches as they cook. Allow the mixture to bubble gently, thickening into a glossy glaze that coats the fruit nicely.
Return the seared chicken breasts to the skillet, nestling them among the peaches. Spoon some of the glaze over the top as the chicken reheats slightly and absorbs the flavors, about 3 minutes.
Sprinkle chopped fresh herbs over the dish and taste to adjust seasoning with salt and pepper. Serve the skillet hot, allowing the caramelized peaches and chicken to shine as a vibrant, saucy centerpiece.