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Whisk together milk, persimmon purée or mash, brown sugar, eggs, raisins, vanilla, and salt in large bowl, stir in bread and let mixture stand at room temperature 15 minutes.
Put oven rack in middle position and preheat oven to 375°.
Stir walnuts into bread pudding, then spoon pudding into a buttered shallow 8-inch square (2-quart) glass or ceramic baking dish, spreading evenly. Dot with butter bits. Bake pudding until golden, puffed, and set, 35 to 40 minutes. Cool to warm in pan on a rack, about 20 minutes.
Bread pudding can be made 1 day ahead and cooled completely, uncovered, then chilled, covered. Reheat, uncovered, in a preheated 350° oven until warm.