Prep the chicken
Dice the onion, mince the garlic, and halve the cherry tomatoes. Tear the basil leaves, keeping a few whole for serving. Pat the chicken breasts dry, butterfly them if needed, and pound them lightly between two sheets of plastic wrap until they are an even thickness. Season both sides with salt, black pepper, and oregano, then set everything aside.
Build the sauce
Heat 2 tbsp of the olive oil in a large skillet over medium heat. Add the onion and cook, stirring, for 5 minutes until soft and translucent. Stir in the garlic and cook for 30 seconds more, just until fragrant, then pour in the dry white wine and let it bubble until the pan is almost dry. Add the cherry tomatoes, a pinch more salt, pepper, and oregano, and simmer, stirring and gently pressing the tomatoes, for 10 minutes until they burst and the sauce turns glossy. Stir in most of the basil and half of the Parmesan, then keep the sauce on low while you finish the chicken. While it simmers, bring a large pot of salted water to a boil for the pasta.
Bread and fry
Set out three shallow dishes with flour, beaten egg, and panko breadcrumbs. Coat each chicken breast in the flour, shake off the excess, dip it in the egg, then press it into the panko so the crumbs cling in an even layer. Heat the remaining 3 tbsp olive oil in a large skillet over medium-high until shimmering, then fry the chicken in batches for 4 minutes per side, until deeply golden and the centers reach 165° F. Move the cutlets to a rack or paper towel-lined plate and let them rest for 2 minutes.
Cook the rigatoni
Bring the pot of water back to a boil if needed, add the Rigatoni, and give it a good stir so the pieces do not clump. Cook until just al dente, about 10 minutes. Before draining, scoop out 1 cup of the pasta water, then drain the rigatoni and return it to the warm pot.
Toss the pasta
Add the rigatoni to the sauce and toss over low heat, loosening with a splash of reserved pasta water if it looks tight. Fold in the torn Buffalo Mozzarella and the remaining basil, tossing just until the cheese softens into creamy pockets. Taste and adjust with a little more salt or pepper if needed.
Plate and serve
Slice the chicken and lay it over the tomato-basil rigatoni, spooning any juices from the plate over the top. Finish with the reserved basil leaves and the remaining Parmesan, then serve right away while the chicken is still crisp.
