Preheat oven to 500° F. Whisk first 3 ingredients in large bowl. Place tomatoes in medium bowl; mix in 2 tablespoons garlic oil, then oregano and crushed red pepper. Add chicken to large bowl with remaining garlic oil and turn to coat.
Place 1 cup Parmesan in pie dish. Dip 1 side of each chicken piece into cheese to coat; arrange chicken, cheese side up, on 1 half of large rimmed baking sheet. Scatter tomatoes on other half of sheet.
Roast chicken until just cooked through and firm to touch, about 10 minutes. Arrange mozzarella slices atop chicken. Return to oven; roast until cheese melts, 1 to 2 minutes. Transfer chicken and tomatoes to 4 plates. Sprinkle with remaining 1/4 cup Parmesan cheese.