How to Make It
In a medium saucepan, heat the pure olive oil until shimmering. Add the onion and rosemary and cook over moderate heat, stirring occasionally, until the onion softens, about 5 minutes. Add the beans, stock, water, carrot and bay leaf and bring to a boil. Cover partially and simmer over moderately low heat until the beans are tender, about 45 minutes. Season the beans with salt and pepper and simmer for 10 minutes longer.
Drain the beans, reserving the cooking liquid. Discard the bay leaf. Transfer half of the beans to a food processor and puree until smooth. Add about 1/4 cup of the reserved cooking liquid and process until creamy. Return the bean puree to the saucepan and stir in the remaining half of the beans; discard the remaining cooking liquid. Season with salt and pepper and keep warm.
Preheat a cast-iron grill pan or the broiler. Brush both sides of the bread with extra-virgin olive oil. Grill or broil until golden brown on both sides. Rub the toasts with the garlic clove, then spread with the bean puree. Arrange the bruschetta on a platter, drizzle with extra-virgin olive oil and sprinkle with sea salt. Serve right away.