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Preheat oven to 400 degrees. Cut chicken into 2-inch pieces. Spread panko on a rimmed baking sheet and bake until golden brown, 6 to 8 minutes. Transfer to a shallow dish, then add Parmesan and 1/2 teaspoon salt; drizzle with oil and stir to combine.
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Place flour and eggs in separate shallow dishes. Increase temperature to 450 degrees.
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Set a wire rack in a rimmed baking sheet; lightly coat rack with cooking spray. In batches, coat chicken in flour, shaking off excess, dip in egg, then coat with panko, pressing to adhere. Place on rack.
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Bake until chicken is cooked through, 12 minutes, flipping halfway through. Serve nuggets with sauce.