Pat the chicken dry and season both sides generously with the Kosher salt and black pepper.
Heat the butter and olive oil in a large high-sided skillet or Dutch oven over medium-high heat until the butter begins to melt and foam.
Add the seasoned chicken and cook undisturbed for 5–7 minutes on the first side, then flip and cook 5–7 minutes more until an instant-read thermometer registers 165°F; transfer the cooked chicken to a plate and set aside.
If needed, add a little more oil to the pan, then add the diced red onion and cook, stirring frequently, until softened, about 3 minutes.
Stir in the Italian seasoning and minced garlic and cook for about 1 minute until fragrant, then add the halved cherry tomatoes and cook briefly to combine.
Sprinkle the flour evenly over the vegetables and cook, stirring, for about 1 minute until the flour is lightly browned.
Slowly pour in the chicken broth while scraping up any browned bits from the bottom of the pan with a wooden spoon.
Add the dry orzo, bring the mixture to a boil, then reduce the heat to low or medium-low to maintain a gentle simmer; cover and cook for about 7 minutes, stirring occasionally, until the orzo is tender and most of the liquid is absorbed.
When the orzo is cooked, fluff and stir, then return the cooked chicken to the pan; add the sliced basil and shredded Parmesan, taste, and adjust seasoning with more salt or pepper if needed.
Serve immediately, garnishing with additional basil or Parmesan if desired.
