Combine the olive oil, vinegar, ½ teaspoon salt, and ¼ teaspoon pepper in a small jar. Put on the lid and shake well to combine.

Cut half a ball of mozzarella into ½-inch chunks and put in a large bowl. Slice the remaining mozzarella into 12 rounds (¼ inch thick) and reserve. Add the tomatoes and olives to the bowl with the mozzarella chunks. Sprinkle with ½ teaspoon salt, drizzle with the balsamic vinaigrette, toss gently, and set aside.

Preheat the broiler. Combine the panko and ½ cup parmesan in a shallow dish. Put the eggs and flour in 2 separate shallow dishes. Season the chicken on both sides with the remaining 1 teaspoon salt and ½ teaspoon pepper. Coat the chicken in the flour, then coat in the eggs, then coat in the panko mixture. Set on a foil-lined baking sheet.

Line a separate baking sheet with foil and top with a wire rack. Heat ½ inch vegetable oil in a large cast-iron skillet over medium-high heat until it starts to shimmer. Add a pinch of panko to the oil; it should sizzle immediately. Working in batches, cook the chicken until golden brown and cooked through, 2 to 3 minutes per side.

Transfer the chicken to the rack and top each with 2 mozzarella slices. Sprinkle with the remaining ¼ cup parmesan. Broil until the cheese is melted, 2 to 5 minutes. Spread each piece of chicken with pesto.

Add the basil to the salad, toss to combine, and season with more salt and pepper, if needed. Serve with the chicken.