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To make the filling: Combine all the ingredients in a small bowl, mixing until smooth. Set aside.
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To make the dough: Break the cookies into the bowl of your food processor and process (using a series of rapid pulses) to make crumbs. Remove a scant 1/2 cup (55g) of the crumbs, place them in a small, wide bowl, and set them aside; you'll dredge the rum balls in them later.
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Add the salt, pecans, and confectioners’ sugar to the remaining crumbs in the processor’s bowl and pulse five or six times to combine. Transfer the mixture to a medium bowl.
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In a microwave-safe measuring cup, combine the rum, brandy, and corn syrup. Microwave for 40 seconds, until just steaming.
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Add the eggnog and vanilla to the warmed liquids and pour all at once into the crumb mixture, stirring until evenly moistened. Let the mixture sit for 5 minutes to hydrate before shaping.
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To shape the rum balls: Using a level tablespoon measure, divide the dough into 24 portions weighing about 20g each.
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Place one portion of dough into the bowl of a tablespoon cookie scoop and make a deep indentation in it with your finger. If you don't have a cookie scoop see "tips," below.
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Place a rounded quarter teaspoon of filling (about 3g) in the indentation, then push the edges over the filling to cover it, pressing down to seal it inside.
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Squeeze the cookie out of the scoop, round it into a ball with your hands, and roll it in the reserved crumbs to coat.
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Repeat until all the portions of dough are filled.
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To make the glaze: In a microwave-safe measuring cup or small bowl, stir the ingredients together until smooth; the mixture will be thick. Heat in the microwave for 30 seconds (it will thin once warmed). Drizzle over the tops of the cookies; the glaze will set as it cools.
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Storage information: Store rum balls in an airtight container at room temperature for up to a week; freeze for up to two months.