Cook the beans: Rinse the beans, place in a saucepan, and add cold water to cover by 2 inches. Bring to a boil, cover, and remove from the heat. Let stand for about 20 minutes, then drain.
Return the beans to the saucepan along with the prosciutto, onion, celery, carrot, garlic, and bay leaf. Pour in the chicken stock and then add water as needed to cover the beans by 2 inches. Slowly bring to a simmer over medium-low heat. (If you heat them too fast, the skins may break.) Adjust the heat to maintain a bare simmer and cook, uncovered, until the beans are almost tender, 20 minutes or longer, depending on the age of the beans. Add salt, to taste, and continue cooking until the beans are tender but not mushy, about 20 minutes longer. Remove from the heat and let the beans cool in the liquid.
Meanwhile, make the bruschetta: Preheat a stove-top grill pan. Place the bread slices in a single layer on the grill pan and toast until each side is crisp and has grill marks, about 6 minutes. (You can also toast the bread in a 500 degrees F oven on a baking sheet.) Remove from the pan and lightly brush the bread slices on both sides with olive oil, then season lightly with salt. Let cool.
Strain the cooled beans and other solids through a sieve, reserving the liquid. In a food processor, puree the beans and solids in batches with 3 1/2 cups of the cooking liquid and 2 tablespoons olive oil. Using a rubber spatula, push the solids through a fine mesh sieve into a clean large saucepan (you should have about 6 cups of soup in the saucepan). Season, to taste, with salt and pepper, then bring to a simmer over medium heat, stirring occasionally.
Meanwhile, in a small skillet over high heat, warm the 1/4 cup olive oil. Turn on the fan above your stove. Add the garlic to the skillet and cook, stirring occasionally, until it begins to turn brown. Add the red pepper flakes and cook for a few seconds, then add the basil. Cook until basil wilts.
To serve, ladle the hot soup into warmed bowls. Top the soup with the basil mixture. Serve bruschetta with each bowl, adding Parmesan, if desired.
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