Melt butter in a large pot over medium heat. Add leek, carrots and celery; season with salt and black pepper. Sauté until softened but not browned, 8-10 minutes.
Add garlic and cook until fragrant, about a minute. Stir in tomato paste and cook until darkened, 2-3 minutes. Add cannellini beans and chicken stock; season with salt and black pepper.
Bring to a boil, reduce to a gentle simmer and cook 15 minutes for flavors to meld. Stir in green beans and kale and simmer until vegetables are cooked, another 10-15 minutes. Add pasta and cook until al dente, about 10 minutes.
Stir in peas and remove from heat. Divide soup among 4 bowls and serve hot.