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To prepare the potatoes: Preheat the oven to 375°F. Prick the potatoes, then place them on a baking sheet and bake until fork-tender, about 1 hour. Set aside to cool.
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To make the dough: In a medium saucepan, melt the butter with the milk over medium heat. Stir in the salt.
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Bring the mixture to a boil, add the flour all at once, and stir until a ball forms.
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Remove the pan from the heat and beat to cool it down a bit.
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Add the eggs one at a time, beating well after each addition. Set aside.
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Peel, then rice or grate the potatoes and measure out 1 cup (140g). Fold the potato into the warm dough along with the herbs.
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To boil the dumplings: In a 3-quart pot, bring 4 cups broth or salted water to a boil. Drop the dough by teaspoonfuls into the simmering liquid, being mindful not to overcrowd the pan; working with about 15 scoops at a time is best.
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Cook the dumplings for 5 to 6 minutes, then remove them from the liquid, transfer them to a heatproof dish, and cover. Keep the dumplings in a warm (200°F) oven while cooking the remaining dumplings.
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Serve warm dumplings as is, with butter; or floating in soup or broth.
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Storage information: Cover and refrigerate any leftover dumplings for up to three days. Reheat the dumplings in a pot of broth set over medium heat, or microwave plain dumplings (without broth) in short bursts until heated through.