Generously season chicken breasts with salt and pepper.
Heat 2 tablespoons olive oil in a large pot. Brown chicken breasts on both sides and continue to cook until cooked through. When cool enough to handle, cut into 1/2 inch pieces; set aside.
Add remaining 2 tablespoons oil and cook onion, garlic, carrots, red bell pepper, and mushrooms and cook until vegetables are soft. Sprinkle vegetables with flour and cook, stirring constantly, for 2-3 minutes.
Remove pot from heat and slowly whisk in broth until smooth.
Return pot to heat, add 5 cups chicken broth, Better Than Bouillon, Italian seasoning, rosemary, bay leaf, 1/4 teaspoon salt, 1/4 teaspoon black pepper, and Parmesan cheese and bring to a boil, then immediately reduce heat and simmer, stirring often so cheese doesn't stick to the bottom of the pot, uncovered, for 20 minutes.
Stir in tortellini adjusting heat to maintain a simmer, continue stirring occasionally. Cook until tortellini is almost al dente. Return chicken, and any juices that have accumulated on the plate, back to the pot and stir in cream. Continue to simmer until chicken is heated through.
Adjust salt and pepper. Remove from the heat and stir in basil. Ladle into individual bowls and serve. As soup sits the pasta will absorb the broth, thin as needed with extra chicken broth.