Heat 2 tablespoons sun-dried tomato oil in a large skillet over medium. Add garlic and crushed red pepper; cook, stirring often, until fragrant, about 1 minute. Add sundried tomatoes, basil, and tomato paste; cook, stirring often, until fragrant and tomatoes are starting to soften, about 2 minutes. Pour in broth; cook until reduced by about half, 2 to 3 minutes. Stir in cream, salt, and pepper, and bring to a simmer over medium. Simmer, stirring occasionally, until sauce is slightly thickened, 3 to 4 minutes. Remove from heat; stir in Parmigiano, stirring constantly until cheese is fully melted. Set aside and cover to keep warm.

Pat chicken cutlets dry and season evenly with salt and pepper. Stir together flour and garlic powder in a wide shallow dish, and set aside. Place egg in a separate shallow dish, and set aside. Stir together panko, Parmigiano, and oregano in a third shallow dish. Working with one cutlet at a time, place chicken in flour, turning to coat, and shake off any excess. Place in egg mixture, turning to coat, and allow any excess to drip off. Place in panko mixture, turning until fully coated, pressing to adhere. Transfer breaded chicken to a plate. Repeat process with remaining chicken.

Fill a large, deep-sided skillet with oil to a depth of 1/4 inch. Heat over medium-high until shimmering. Working in two batches, carefully lay breaded cutlets in skillet. Cook, flipping once, until golden and crispy, 2 to 3 minutes per side. Transfer to a paper-towel lined plate to drain. Repeat with remaining chicken. 

Place chicken cutlets on a rimmed baking sheet lined with aluminum foil. Spread 2 teaspoons Sauce over each cutlet, then top evenly with sliced mozzarella. Broil in preheated oven until cheese is melted and golden, and a thermometer inserted into thickest part of chicken registers 165°F, 3 to 4 minutes.

To serve, spread remaining Sauce over a serving platter. Transfer cutlets to platter over the sauce, garnish with basil, and serve immediately.