Step 1: Carefully separate the yolks from the whites of 6 large eggs. Place the yolks in a large heatproof bowl and the whites in a separate, clean bowl (these can be reserved for another use or whipped and folded in later).
Step 2: Add 3/4 cup granulated sugar to the bowl with the egg yolks. Whisk vigorously until the mixture is pale yellow and thick, forming ribbons when the whisk is lifted.
Step 3: In a medium saucepan, combine 3 cups whole milk, 1 cup heavy cream, 1/2 teaspoon freshly grated nutmeg, and a pinch of salt. Heat over medium heat, stirring occasionally, until the mixture is simmering gently around the edges but not boiling.
Step 4: Slowly pour about half of the hot milk mixture into the whisked egg yolks, whisking constantly and vigorously. This tempering process prevents the yolks from scrambling. Once combined, pour the tempered yolk mixture back into the saucepan with the remaining hot milk mixture.
Step 5: Continue to cook the mixture over medium-low heat, stirring constantly with a wooden spoon or spatula, until it thickens enough to coat the back of the spoon. It should reach about 160-170°F (71-77°C) on an instant-read thermometer. Do not let it boil!
Step 6: Remove the saucepan from the heat and stir in 1 teaspoon pure vanilla extract. Pour the egg nog base through a fine-mesh sieve into a clean bowl or pitcher to remove any potential lumps or spices. Cover and refrigerate for at least 4 hours, or preferably overnight, until thoroughly chilled. This allows the flavors to meld beautifully.
Step 7: If using, slowly stir in 1 1/2 cups of your chosen spirit (brandy, dark rum, or bourbon) into the thoroughly chilled egg nog base. If desired for a lighter texture, whip the reserved egg whites (or 1 cup of additional heavy cream) to soft peaks and gently fold them into the chilled mixture just before serving.
Step 8: Pour the homemade egg nog into individual serving glasses. Garnish each serving with an extra sprinkle of freshly grated nutmeg and, if desired, a cinnamon stick.