In a large bowl, whisk together the whole milk, heavy cream, granulated sugar, eggnog, and peppermint schnapps until the mixture is completely smooth.
Add the vanilla extract and grenadine to the mixture, stirring gently to incorporate.
Taste the mixture and adjust sweetness if needed, adding more grenadine or sugar to suit your preference.
Transfer the punch into a large serving bowl or pitcher, and chill in the refrigerator for at least one hour to allow the flavors to develop.
Just before serving, add crushed ice to the punch for a refreshing chill. Stir gently to combine.
Serve the punch in glasses, topped with a dollop of whipped cream and a peppermint stick for garnish.