In a saucepan, combine milk, cardamom, and cinnamon. Using a vegetable peeler, peel a strip of zest from the orange, avoiding the white pith as much as possible. (You can scrape any white bits off with a knife.) Using a microplane or grater, zest orange to yield about 1 teaspoon.
Add the strip of zest to the milk and heat the milk to a simmer. Do not let it boil! Stir to keep the milk from scorching, and once bubbles appear at the edge of the pan, cover and remove from heat.
Place the chocolate in a heatproof bowl - one with a spout for pouring works well for this, or a 4-cup measuring cup.
Set a strainer over the bowl, and pour the milk through the strainer into the chocolate. Add nutmeg and vanilla. Carefully whisk mixture until smooth.
Pour into mugs and top with whipped cream, grated orange zest, and a pinch of nutmeg.