Grate the fresh ginger finely using a microplane or fine grater, aiming for about 1 tablespoon of zest and pulp.
Combine the milk, molasses, and grated ginger in a small saucepan and place over low heat.
Add the ground cinnamon, nutmeg, cloves, and vanilla extract to the saucepan, stirring gently to distribute the spices evenly.
Warm the mixture gently, stirring often, until it is steaming and fragrant, about 5-7 minutes. Do not let it boil.
Once steaming, remove the saucepan from heat and strain the mixture through a fine mesh sieve into a clean bowl or jug, pressing gently to extract all the aromatic liquid.
Pour the hot, strained spice milk into mugs, filling each about three-quarters full.
If desired, top with whipped cream and a sprinkle of cinnamon or crushed ginger snap cookies for extra flavor and presentation.
Serve immediately while warm, inhaling the rich, spicy aroma with each sip.