Pat chicken dry and season both sides with 1 teaspoon kosher salt and 1 teaspoon freshly ground black pepper.
Heat a large high-sided skillet or Dutch oven over medium-high heat; add 2 tablespoons butter and 1 tablespoon olive oil and warm until butter begins to melt.
Add the chicken and sear undisturbed 5–7 minutes on the first side, flip, and cook another 5–7 minutes until an instant-read thermometer reads 165°F; transfer chicken to a plate and set aside.
If needed, add a little more oil to the pan, then add 1/3 cup finely diced red onion and cook about 3 minutes until softened, stirring frequently.
Add 1 teaspoon Italian seasoning, 3–4 minced garlic cloves, and 1½ cups halved cherry tomatoes; cook about 1 minute until fragrant, stirring constantly.
Sprinkle 1 tablespoon all-purpose flour evenly over the vegetables and cook about 1 minute until the flour is lightly browned.
Slowly pour in 3 cups chicken broth, stirring and scraping the browned bits from the pan with a wooden spoon.
Add 1 cup dry orzo, bring the mixture to a boil, then reduce heat to low to maintain a gentle simmer; cover and cook about 7–10 minutes until most liquid is absorbed and orzo is tender, stirring occasionally.
Fluff the orzo, return the cooked chicken to the pan, then stir in 1/2 cup thinly sliced fresh basil and 1/2 cup shredded Parmesan; taste and adjust seasoning as needed.
Serve immediately while hot.