To make the croutons, heat the olive oil over medium heat in a large frying pan. Add the garlic and saute for 1 minute. Remove from the heat and let stand for 10 minutes to infuse the garlic flavor into the oil. Remove the garlic pieces and discard. Return the pan to medium heat. Add the bread cubes and saute, stirring frequently, until lightly browned, 3 to 5 minutes. Transfer to a small bowl and set aside.
In a soup pot over high heat, combine the white beans, water, 1/2 teaspoon of the salt and the bay leaf. Bring to a boil over high heat. Reduce the heat to low, cover partially and simmer until the beans are tender, 60 to 75 minutes. Drain the beans, reserving 1/2 cup of the cooking liquid. Discard the bay leaf. Place the cooked beans into a large bowl and save the cooking pot for later use.
In a small bowl, combine the reserved cooking liquid and 1/2 cup of the cooked beans. Mash with a fork to form a paste. Stir the bean paste into the cooked beans.
Return the cooking pot to the stove top and add the olive oil. Heat over medium-high heat. Stir in the onion and carrots and saute until the carrots are tender-crisp, 6 to 7 minutes. Stir in the garlic and cook until softened, about 1 minute. Stir in the remaining 1/2 teaspoon salt, the pepper, chopped rosemary, bean mixture and stock. Bring to a boil, then reduce the heat to low and simmer until the stew is heated through, about 5 minutes.
Ladle the stew into warmed bowls and sprinkle with the croutons. Garnish each bowl with a rosemary sprig and serve immediately.