Beans:
Soup:
To make the beans: Pick the beans over and soak for 4 to 6 hours. Drain the beans and place them in a large pot. Add the carrot, celery, onion, garlic, and parsley stem sachet. Pour in 3 quarts cold water and bring to a boil, skimming off any foam that rises to the top. Decrease the heat to a gentle simmer and cook until the beans are soft and creamy but not falling apart, 30 to 90 minutes. (Start checking after 25 minutes; the fresher the beans, the shorter the cooking time.) Add the salt, pepper, and olive oil.
Remove the pot from the heat and discard the sachet and vegetables. Let the beans cool in their broth.
To make the soup: Peel and finely grate the garlic cloves, then set aside. Trim off the dark green leek tops, then halve the white and light green parts lengthwise. Rinse, then finely chop. Warm the olive oil in a large pot over medium heat. Add the grated garlic, leeks, carrots, celery, chile flakes, and rosemary. Cook, stirring occasionally, until the rawness of the vegetables is just gone, 3 to 4 minutes. Add the tomatoes and cook until slightly caramelized, about 3 minutes. Add the beans and their cooking broth and bring to a boil. Decrease the heat and simmer until the vegetables are tender, 5 to 10 minutes. Season to taste with salt and pepper.
Just before serving, add the parsley. Toast the bread slices. While still hot, rub them with the garlic halves, then tear into large pieces. Ladle the soup into bowls and top each with a few toasts, then drizzle with olive oil.
