Cut ham off bone in bite-size pieces, reserving bone. Melt 1 tablespoon butter in a skillet over medium heat and fry ham pieces in hot butter until lightly browned, about 4 minutes.
Melt 1 tablespoon butter in a large saucepan; cook and stir onion, carrots, and garlic in the hot butter until softened and onion is translucent, about 5 minutes. Pour undrained beans into vegetables, add ham and reserved ham bone, and pour in water to make soup your desired thickness. Stir bay leaves into soup and season with salt and black pepper.
Bring soup to a boil, reduce heat to medium-low, and simmer for 25 minutes. Remove ham bone and bay leaves before serving.
