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lightly season chicken cutlets all over with salt and pepper; let rest while you make the basil oil.
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in a blender or food processor, combine oil, basil, garlic and salt to taste; purée until smooth.
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in a medium bowl, toss tomatoes and mozzarella with about half of the basil oil and salt to taste.
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set aside.
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place flour in a shallow bowl or plate.
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pour eggs into another shallow dish; combine panko and parmesan in a third dish.
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in a large skillet, melt oil and butter over medium heat.
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as it heats, dip a chicken cutlet in flour, shake off excess, then dip in eggs.
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shake off excess, then dip both sides in panko mixture and transfer to a rimmed baking sheet.
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repeat with remaining cutlets.
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when oil is hot, fry 2 cutlets at a time until bottoms are golden, about 3 minutes.
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flip and fry until golden and crispy all over, about another 3 minutes.
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transfer to a paper towel-lined plate.
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sprinkle immediately with salt.
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taste tomato-mozzarella salad and add more salt and/or basil oil if needed.
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serve cutlets topped with a squeeze of lemon, a drizzle of remaining basil oil and tomato-mozzarella salad.