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1
Blend 3 cups flour, 1/4 cup white sugar, yeast, and salt in a food processor. Add warm milk, eggs, and 1/4 cup softened butter and pulse until dough forms into a ball. Transfer dough to a bowl and allow to sit at room temperature to rise, about 1 hour. Knead dough and roll onto a work surface. Cut dough into 8 circles using a cookie cutter or the rim of a drinking glass; allow to rise, about 30 minutes.
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2
Melt 3 tablespoons butter in a heavy pot over medium heat; cook and stir 1/4 cup sugar into butter until browned, about 5 minutes. Add 1 cup milk to butter-sugar mixture. Arrange dough circles in the pot; cover and cook over low heat until dough is cooked through, 10 to 15 minutes.
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3
Melt 2 tablespoons unsalted butter in a small saucepan over medium heat; stir 2 tablespoons flour into butter until mixture bubbles, 1 to 2 minutes. Gradually pour 1 cup milk into butter-flour mixture, cooking and stirring until thickened, 3 to 5 minutes. Stir 1/2 cup sugar, brandy, and 1 teaspoon vanilla extract into thickened milk mixture until a smooth vanilla sauce forms, 2 to 3 minutes.
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4
Mix 2 cups sugar and 1/4 cup warm water together in a saucepan over medium-low heat until sugar is dissolved, about 5 minutes. Very gradually add remaining water, stirring constantly, until mixture is smooth, 2 to 4 minutes. Mix maple syrup, cream, 1 tablespoon unsalted butter, and 1 dash vanilla extract into sugar mixture until a thick caramel sauce forms, about 5 minutes.
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5
Transfer dampfnudel to a serving dish and pour vanilla sauce over each; top with caramel sauce.
