In a medium bowl, whisk together egg yolks and 1/2 cup sugar until light and creamy, about 2 minutes.
In a medium saucepan, combine milk, heavy cream, nutmeg, and cinnamon. Heat over medium heat until it just begins to simmer, stirring occasionally.
Slowly pour about 1 cup of the hot milk mixture into the egg yolk mixture while whisking constantly to temper the eggs. Then pour the egg mixture back into the saucepan.
Cook over medium-low heat, stirring constantly, until the mixture thickens slightly and coats the back of a spoon (about 160°F), approximately 5-8 minutes. Do not let it boil.
Remove from heat and stir in vanilla extract. Strain through a fine-mesh sieve into a large bowl and let cool to room temperature. Refrigerate for at least 2 hours or overnight.
Just before serving, beat egg whites with salt until soft peaks form. Gradually add remaining 1/4 cup sugar and beat until stiff peaks form.
Gently fold the beaten egg whites into the chilled eggnog mixture. Serve immediately with a sprinkle of freshly grated nutmeg on top.