1 1/4 cups white sugar
1 cup dark rum
1 cup bourbon
1 cup brandy
1 pinch salt
1 teaspoon vanilla extract
1 pinch ground nutmeg, or to taste
1 quart half-and-half
1 pint heavy whipping cream
Mix sugar, rum, bourbon, brandy, salt, vanilla extract, and nutmeg in a blender on medium speed until combined. Transfer liquor mixture to a large bowl.
Mix half-and-half, cream, and eggs in the blender on low speed until combined. Stir egg-cream mixture into the liquor mixture.
Ladle eggnog mixture into the blender in batches; blend on low speed. Transfer blended eggnog to a punch bowl or pitcher. Refrigerate until cold, 1 to 2 hours.