Cut lamb into 1-inch chunks.
In a 5- to 6-quart pan, combine lamb, thyme, and 1 cup water. Cover and bring to a boil over high heat. Reduce heat and simmer for 20 minutes.
Uncover and boil over high heat, stirring often, until a brown film forms on pan bottom, about 15 minutes. Add 1/3 cup broth and stir to loosen browned bits, then stir often until lamb is lightly browned, about 4 minutes.
Add remaining broth and bay leaves; bring to a boil.
Cut 1/4 of the top off the garlic head. Add garlic head to broth. Cover, reduce heat, and simmer until lamb is tender when pierced, about 1 hour.
With a slotted spoon, lift out garlic and set aside. To pan, add tomatoes (including juice), sausages, and beans.
Squeeze garlic cloves from the head and add cloves to soup. Reduce heat, cover, and simmer to blend flavors, about 5 minutes, stirring occasionally.
If soup is thicker than you like, thin with a little water and simmer 2 to 3 more minutes to blend flavors. Ladle into bowls or mugs and add salt and pepper to taste.