1. Prepare the Chicken: If desired, quickly sear the chicken breasts or thighs in a skillet with 1 tablespoon of olive oil for 2-3 minutes per side before adding them to the slow cooker. Place the chicken pieces evenly at the bottom of your slow cooker.

2. Combine the Tomato Base: In a medium bowl, whisk together the crushed tomatoes, diced tomatoes (undrained), chicken broth, minced garlic, dried oregano, salt, and black pepper. Stir until all the ingredients are well combined.

3. Pour Over Chicken: Carefully pour the tomato mixture over the chicken in the slow cooker, ensuring the chicken is mostly submerged in the sauce.

4. Slow Cook: Cover the slow cooker with its lid. Cook on the LOW setting for 6-8 hours, or on the HIGH setting for 3-4 hours. The chicken is done when it is tender and easily shredded with a fork.

5. Shred the Chicken: Once cooked, carefully remove the chicken from the slow cooker and place it on a cutting board. Using two forks, shred the chicken into bite-sized pieces. Return the shredded chicken to the slow cooker.

6. Add Cream and Basil: Stir in the heavy cream and fresh chopped basil into the slow cooker with the shredded chicken and tomato sauce. Stir gently until everything is well combined and heated through, about 10-15 minutes on low or warm setting.

7. Taste and Adjust: Taste the sauce and adjust seasonings if necessary. You might want to add a pinch more salt, pepper, or even a dash of red pepper flakes for a little heat.

8. Serve Warm: Ladle the creamy tomato basil chicken over your favorite cooked pasta, rice, or serve with crusty bread for dipping. Garnish with fresh parsley or extra basil, if desired.