Heat olive oil in a large skillet over medium-high heat. Add chicken to the skillet and season with salt and pepper. Sauté until chicken is cooked through and has browned, 5 to 10 minutes. Remove from skillet and transfer to a bowl; cover to keep warm. Set aside.

Deglaze pan with chicken broth over medium heat, scraping up any browned bits. Stir in pesto and heavy cream. Cook until thickened, 3 to 5 minutes. Mix in 1/2 cup Parmesan cheese and stir until melted. Add in spinach and tomatoes. Cook until spinach begins to wilt, 3 to 5 minutes.

Add tortellini and cooked chicken and continue cooking until bubbly and hot, 5 to 10 minutes. Sprinkle with remaining 1/2 cup Parmesan cheese and serve.