Bring a large pot of water to a boil. Season very generously with salt, bring back up to a boil, then add the pasta to cook according to package instructions, or until al dente. Before draining, reserve 1 cup pasta water. Drain and toss the pasta lightly in oil to prevent any sticking.
In a large skillet over medium high heat, add the olive oil until glistening, about 1 minute.
Add the diced tomatoes, shallots, and garlic. Season with then salt. Saute for 6 to 8 minutes, until tomatoes are cooked down and very tender.
Add the vodka to deglaze the skillet, stirring until mostly evaporated, about 3 to 4 minutes.
Transfer the tomatoes to a high speed blender (a small compact blender or immersion blender is best!) Blend until very smooth, adding a little reserve pasta water as needed to help loosen up the mixture. Add back into the skillet. Reduce heat to medium.
Stir in the heavy cream, grated parmesan, dried red pepper flakes, and dried oregano. Add the pasta and toss for a good minute or so to coat the pasta in the sauce. Add the pasta water 1/4 cup at a time to help loosen the sauce and adhere better to the pasta.
Enjoy warm with the parmesan chicken cutlets and freshly grated parmesan and chopped basil to serve.