For a quick infusion use ground ginger instead of ginger root: one teaspoon of ground ginger to a cup and a half of brandy or rum (if you use rum the name of the cocktail makes more sense). Allow to sit for 15-30 minutes and strain through a coffee filter.
Combine ingredients with ice and shake until frothy
Serve up in a martini glass
Garnish with nutmeg
Ginger Eggnog Martini Cocktail https://www.recipefiction.com/2014/12/ginger-eggnog-martini-why-you-should-infuse-your-own-liquor.html