Heat 1 tablespoon olive oil in a large skillet over medium-high heat until shimmering.
Add the cubed chicken, season with salt and pepper, and cook 4–5 minutes, stirring occasionally, until browned and cooked through; transfer chicken to a plate.
Reduce heat to medium, add 2 tablespoons butter to the same pan and melt.
Whisk in 2 tablespoons flour and cook 1 minute to form a roux.
Slowly whisk in 2½ cups chicken broth and cook 2 minutes, stirring, until slightly thickened.
Add 20 ounces refrigerated tortellini and bring the mixture to a boil; cook about 4 minutes, stirring occasionally so tortellini is coated and heated through.
Stir in ½ cup heavy cream and ¼ cup grated Parmesan and cook 1 minute more until sauce is smooth.
Return the cooked chicken to the pan and add 1½ cups fresh spinach; cook 2–3 minutes until spinach is wilted.
Fold in 1 cup halved cherry tomatoes, adjust seasoning if needed, then sprinkle with chopped parsley and serve.
