Drain the dried beans and place in large saucepan with cold water to cover. Bring slowly to a boil, drain and rinse. Rinse out the saucepan, return the beans to the pan, and add cold water to cover by 3 inches. Bring to a boil, reduce the heat, cover, and gently simmer 25 to 30 minutes, or until soft. Drain, reserving 1/2 cup of the cooking liquid. Remove the fava bean skins with your fingers and spread the beans between layers of paper towels to dry.
Heat 2 tablespoons of the olive oil in a large heavy skillet. Saute the onion, garlic, carrot, celery, and bay leaf over medium-low heat for 15 to 20 minutes, or until dark golden brown, stirring occasionally with a wooden spoon.
Transfer to a food processor or blender container and add the bread.
With the machine running, add about two thirds of the lemon juice and 2 tablespoons of the olive oil, and process until thick and smooth. Add a few tablespoons of water if the puree seems too thick, and add salt, pepper, olive oil, and/or lemon juice to taste.
Serve on a platter, sprinkled with the remaining parsley and olive oil and the paprika.
To make the traditional way: Pound the cooked vegetable-bean mixture and the bread in a large wooden mortar for 5 minutes. Slowly add about two thirds of the lemon juice and then 3/4 cup of the olive oil. Pound for 5 minutes longer, and add salt, pepper, olive oil, and/or lemon juice to taste.