Heat 1 tablespoon olive oil in a large pan over medium-high heat. Add the 1 pound cubed chicken breasts, season with salt and pepper, and cook 4–5 minutes, stirring occasionally, until the chicken is browned and just cooked through. Transfer the chicken to a plate and set aside.
Reduce heat to medium. Add 2 tablespoons unsalted butter to the pan and melt.
Whisk in 2 tablespoons all-purpose flour and cook, whisking constantly, for 1 minute to form a roux.
Gradually whisk in 2½ cups chicken broth, scraping up any browned bits from the bottom of the pan. Bring to a simmer and cook about 2 minutes, until the liquid is slightly thickened.
Add 20 ounces refrigerated tortellini to the pan and bring the mixture to a boil. Cook for about 4 minutes, stirring occasionally so the tortellini cooks evenly and stays coated with the liquid.
Stir in ½ cup heavy cream and ¼ cup grated Parmesan cheese. Cook 1 minute more, stirring, until the sauce is smooth and slightly thickened.
Return the cooked chicken to the pan and add 1½ cups fresh spinach. Cook 2–3 minutes, stirring occasionally, until the spinach is wilted and the chicken is heated through.
Fold in 1 cup halved cherry tomatoes. Taste and adjust seasoning with salt and pepper if needed.
Remove from heat, sprinkle with chopped parsley, and serve.