Wash the split peas thoroughly under cold running water and continue to wash until the draining water runs clear. Pick over the peas and discard any discolored ones. In a heavy 3 to 4 qt. Suacepan, bring the 2 cups of water to a boil and drop in the peas slowly so that the water continues to boil. Reduce the heat and simmer partially covered for 1 1/2 hours, or until the peas can be easily mashed aganst the side of the pan with a spoon. Drain the peas in a colander and puree them in a food mill or force them through a fine sieve set over a large bowl. Return the peas to the pan and cook over low heat, stirring constantly, until the puree is heated through. Stir in the salt, butter, and pepper, and taste for seasoning. Serve at once from a heated vegetable dish. Pease pudding is the traditional accompaniment to boiled beef and carrots with dumplings (see recipe that follows this one.)