Pat chicken dry and season evenly with 1 teaspoon of the salt. Heat oil from sun-dried tomatoes in a large Dutch oven over medium until shimmering. Add chicken, top-side down; cook, undisturbed until browned, about 4 minutes. Turn and continue to cook until browned on second side, about 3 more minutes. Transfer chicken to a plate and set aside (chicken will not be cooked through).

Add onion, celery, and carrots to oil in Dutch oven and cook over medium, stirring occasionally, until softened, about 8 minutes. Add tomato paste and sundried tomatoes and cook, stirring constantly, until deeper in color, about 3 minutes. Add garlic and red pepper flakes; cook stirring constantly, until fragrant, about 1 minute.

Add reserved chicken, stock, and Parmesan rind, if using, and bring to a boil over high. Once boiling, reduce heat to medium; simmer until an instant read thermometer registers 165°F when inserted into chicken thighs, about 8 minutes. Remove chicken to a cutting board and shred into bite-sized pieces.

Return mixture to a boil, add noodles and cook over high, stirring occasionally, until al-dente, about 9 minutes, stirring in remaining ½ teaspoon salt, spinach and shredded chicken during last minute of cooking. Remove and discard Parmesan rind. Remove from heat and stir in cream, ½ cup of the Parmesan, and cream cheese until melted. Divide among bowls and top evenly with remaining ¼ cup Parmesan and basil.