Rinse all of the vegetables well, including the kombu. In a 12-quart or larger stockpot, combine the carrots, onions, leek, celery, potatoes, sweet potatoes, yam, garlic, parsley, kombu, peppercorns, allspice berries, and bay leaves. Fill the pot with the water to 2 inches below the rim, cover, and bring to a boil.

Remove the lid, decrease the heat to low, and simmer, uncovered, for at least 2 hours. As the broth simmers, some of the water will evaporate; add more if the vegetables begin to peek out. Simmer until the full richness of the vegetables can be tasted.

Strain the broth through a large, coarse-mesh sieve (remember to use a heat-resistant container underneath), then add salt to taste.