Bring a large pot of salted water to a boil. Cook the tortellini according to package instructions, adding the broccoli florets during the last 3 minutes of cooking time. Drain and set aside.

In a large skillet or Dutch oven, heat 2 tablespoons of olive oil over medium-high heat. Season the chicken pieces with salt and pepper, then add to the hot skillet. Cook until golden brown and cooked through, about 6-7 minutes. Transfer to a plate and set aside.

In the same skillet, add the remaining tablespoon of olive oil and the butter. Once the butter has melted, add the minced garlic and cook until fragrant, about 30 seconds, being careful not to burn it.

Sprinkle the flour over the garlic and butter, whisking constantly for about 1 minute to create a roux.

Slowly pour in the half-and-half and chicken broth, whisking continuously to prevent lumps from forming. Bring the mixture to a simmer.

Reduce heat to medium-low and add the grated Parmesan cheese, dried basil, dried oregano, and red pepper flakes (if using). Stir until the cheese has melted and the sauce is smooth.

Continue to simmer the sauce, stirring occasionally, until it thickens enough to coat the back of a spoon, about 5-7 minutes.

Return the cooked chicken to the skillet, stirring to coat with the sauce. Taste and adjust seasoning with salt and pepper as needed.

Add the drained tortellini and broccoli to the skillet, gently folding everything together until well coated with the creamy sauce. If the sauce seems too thick, add a splash of the reserved pasta cooking water to loosen it up.

Remove from heat and let stand for 5 minutes to allow the pasta to absorb some of the sauce.

Serve hot, garnished with fresh parsley and additional grated Parmesan cheese.