Dice the onion into 1/2-inch pieces, cut the carrot into 1/4-inch rounds or half-moons, chop the celery into 1/4-inch pieces, and mince the garlic cloves. Cut the ham into 1-inch chunks and roughly chop the kale, removing any thick stems. Having everything prepped and ready will make the cooking process smooth and ensure your aromatics don't overcook while you're still chopping.
Heat the butter in a large pot over medium heat until it foams. Add the diced onion and cook for 4-5 minutes, stirring occasionally, until it becomes soft and translucent. Then add the carrot and celery, cooking for another 4-5 minutes until they start to soften. Finally, add the minced garlic and cook for about 1 minute until fragrant. This layered approach allows each vegetable to develop properly and builds a flavorful foundation for the soup.
Add the ham chunks to the pot, stirring to combine with the vegetables. Pour in the chicken broth, then add the rinsed and drained cannellini beans. Sprinkle in the thyme, bay leaf, black pepper, smoked paprika, and salt. Stir everything together until well combined. The rinsing of the beans removes excess starch, which helps keep the soup broth clear rather than cloudy.
Bring the soup to a boil over medium-high heat, then reduce to medium-low and cover with a lid. Simmer for 25 minutes, allowing the flavors to meld and deepen. I like to use a lower temperature here because it prevents the beans from breaking apart, which keeps the soup looking nice and hearty. Check partway through that nothing is sticking to the bottom and stir if needed.
If you prefer a thicker, creamier soup, carefully ladle about 2 cups of the soup into a blender and blend until smooth (be cautious with hot liquids—let it cool slightly first or use a kitchen towel over the lid). Return the blended mixture to the pot and stir to combine. For a brothier texture, skip this step entirely. Blending just a portion creates a nice middle ground between chunky and creamy.
Add the chopped kale to the pot and simmer uncovered for 5 minutes until the kale is tender but still vibrant green. Taste the soup and adjust the seasoning with additional salt and pepper as needed. Remember that the ham and broth already contribute saltiness, so taste before adding more.
Ladle the soup into bowls and serve immediately while hot. This hearty soup pairs wonderfully with crusty bread, a simple green salad, or cornbread on the side.