Cook the pasta in a large pot of salted water according to package directions until al dente. Drain, reserving ½ cup of pasta water.
Season the chicken with salt and pepper, then dredge lightly in flour.
In a large skillet, heat the butter and oil over medium-high heat. Brown the chicken for 4–5 minutes per side until golden. Remove chicken and set aside.
In the same skillet, add the garlic and sun-dried tomatoes. Sauté for 1 minute.
Stir in the oregano, red pepper flakes, chicken broth, and heavy cream. Simmer for 3–5 minutes until the sauce begins to thicken.
Stir in the Parmesan cheese until melted and smooth.
Return the chicken to the skillet and simmer for another 2 minutes.
Add the cooked pasta to the skillet, tossing to coat. If the sauce is too thick, add a splash of the reserved pasta water.
Top with fresh basil and serve immediately.