In a medium bowl, beat the sugar into the eggs until thickened and pale yellow. Beat in the flour and salt. Set aside.
Bring the milk to a simmer in a heavy medium saucepan. Slowly beat the hot milk into the eggs and sugar mixture. Pour the entire mixture back into the saucepan and place over low heat. Stir constantly with a wooden spoon until the custard thickens slightly and coats the back of the spoon. Be careful not to let the mixture boil or the eggs will scramble. Remove from heat and pour the custard through a strainer into a large clean bowl. Allow the custard to cool slightly, and stir in the eggnog and vanilla. Cover and refrigerate until cold or overnight.
Stir the chilled custard, and freeze in your ice cream machine according to the manufacturer’s directions.