In a large, high-sided skillet over medium heat, heat 4 Tbsp. oil. Add Italian seasoning and cook, stirring frequently, until fragrant, about 10 seconds. Add chicken and cook, stirring, until heated through, about 2 minutes; season with 1 tsp. salt. Transfer chicken mixture to a plate.
In same skillet over medium heat, heat remaining 2 Tbsp. oil. Cook onion and chile, scraping up browned bits on bottom of skillet, until onions are soft and translucent, about 7 minutes; season with remaining ½ tsp. salt. Add garlic and cook, stirring, until fragrant, about 1 minute. Add tomato paste and cook, stirring to coat vegetables, until just starting to caramelize, about 1 minute. Add tomatoes and cook, stirring frequently, until softened and skins wilted, about 5 minutes.
Add broth and beans, return chicken and any accumulated juices to skillet, and bring to a simmer. Add cream and spinach and cook, stirring, until spinach is wilted, 3 to 4 minutes. Stir in cheese and lemon zest until cheese is melted.