In a large heavy pot, cook the bacon over medium heat until the fat is rendered and the bacon is crisp, about 7 minutes. Remove the bacon pieces from the pot and drain on paper towels. Reserve the bacon for garnish.
Add the onions, carrots, celery, bell pepper, garlic, and bay leaves to the bacon fat. Turn the heat to medium-low and cook the vegetables, stirring frequently, for 10 minutes. Add the sausage to the pot and cook until the sausage is lightly browned, about 5 more minutes.
Stir in the thyme, navy beans, and kidney beans. Add the tomatoes, crushing the tomatoes with your fingers as you add them to the pot. Add the chicken broth, and season with salt and pepper to taste.
Bring the mixture to a boil over high heat. Reduce the heat to low, cover the pot, and simmer for 30 minutes. Discard the bay leaves and thyme sprig. Season with additional salt and pepper, if necessary.
Ladle into bowls and top with the reserved bacon, grated cheese, and additional fresh herbs.