Crack the eggs into the jar of a high-powered blender and pour in the maple syrup. Secure the lid tightly, then blend on the highest speed for exactly 4 minutes, using a timer. The mixture should become lighter in color, slightly thickened, and a bit frothy on top.
Add the milk, heavy cream, brandy, rum, and nutmeg to the blender. Blend again for about 10 seconds, just long enough for everything to come together into a smooth, unified mixture without over-foaming.
Carefully pour the eggnog into an airtight glass bottle or container. Place it in the refrigerator and chill until very cold, allowing the flavors to meld and the texture to become luxuriously creamy.
When ready to serve, give the eggnog a good stir or shake to redistribute any settled spices. Pour into glasses and finish with a cinnamon stick or a light dusting of ground cinnamon on top for a warm, festive touch.