Heat the olive oil in a large pot over medium heat. Add the diced onion and minced garlic, then cook for 3 to 4 minutes until the onion softens.
Place the chicken breasts in the pot. Sprinkle with Italian seasoning, salt, and pepper. Cook for 5 to 6 minutes per side until lightly browned.
Pour in the chicken broth. Add the white beans and sun-dried tomatoes. Bring the mixture to a simmer and cook for 15 minutes.
Remove the chicken breasts, shred them with two forks, and return the pieces to the pot.
Stir in the chopped kale and let it wilt for 3 minutes. Reduce the heat to low, then slowly add the heavy cream and Parmesan cheese while stirring until the soup thickens slightly. Adjust salt and pepper as needed before serving.