Remove tenderloins from chicken breasts for another use. Slice each breast horizontally into 2 thin cutlets. Cover cutlets with plastic wrap and pound to ¼-inch thickness.
Beat eggs, 1 teaspoon salt, and pepper in a shallow dish. In a second dish, combine panko and Parmesan. Dip each chicken cutlet in the egg mixture, let excess drip off, then coat with the panko mixture, pressing gently to adhere.
Preheat the oven to 200°F. Heat ¼ cup vegetable oil in a 12-inch skillet over medium-high heat. Cook 4 cutlets until golden brown on both sides, 2–3 minutes per side. Transfer to a paper towel-lined platter, season with salt, and keep warm in the oven. Repeat with the remaining oil and cutlets.
Whisk vinegar, shallot, mustard, and ¼ teaspoon salt in a large bowl. Gradually whisk in 3 tablespoons of olive oil until emulsified.
Heat the remaining 1 tablespoon olive oil in the skillet over medium heat. Cook fennel until softened and lightly browned, 5–7 minutes. Transfer fennel to the vinaigrette.
Toss the warm fennel with cherry tomatoes, radicchio, and arugula in the vinaigrette. Adjust seasoning with salt and pepper.
Plate the crispy chicken cutlets alongside the salad and enjoy!