Pat chicken dry and season both sides generously with kosher salt and freshly ground black pepper.
Heat a large high-sided skillet or Dutch oven over medium-high heat and add the butter and olive oil; melt and swirl to coat.
Add the chicken and sear without moving for 5–7 minutes on the first side, then flip and cook another 5–7 minutes until an instant-read thermometer registers 165°F; transfer chicken to a plate.
If needed, add a little more olive oil to the pan, then add the diced red onion and cook, stirring frequently, until softened, about 3 minutes.
Stir in the Italian seasoning, minced garlic, and halved cherry tomatoes and cook about 1 minute until fragrant.
Sprinkle the tablespoon of flour evenly over the vegetables and cook, stirring, for about 1 minute until lightly browned to remove the raw flour taste.
Slowly pour in the chicken broth while scraping up browned bits from the pan with a wooden spoon.
Add the dry orzo, bring the mixture to a boil, then reduce heat to low or medium-low to maintain a gentle simmer; cover and cook about 7 minutes, stirring occasionally, until orzo is tender and most liquid is absorbed (cook up to 10 minutes if needed).
Fluff the orzo, return the cooked chicken to the pan, stir in the basil and shredded Parmesan, and taste; adjust seasoning with more salt, pepper, or Italian seasoning as desired.
Serve immediately.