Make an ice bath by putting some ice and a little water in a large bowl and nest a smaller metal bowl (one that will hold at least 2 quarts) inside it. Set a mesh strainer over the top.
In a medium saucepan mix the milk, cream, granulated sugar, ginger, ground cinnamon, cinnamon stick, nutmeg, and salt.
Warm the mixture until hot and the edges begin to bubble and foam.
Whisk the egg yolks in a separate bowl and gradually whisk in about half of the warm spiced milk mixture, stirring constantly.
Scrape the warmed yolks back in to the saucepan and cook over low heat, stirring constantly and scraping the bottom with a heatproof spatula, until the mixture thickens and coats the spatula. If using an instant-read thermometer, it should read between 160º-170ºF (71º-76ºC).
Immediately pour the mixture through the strainer into the bowl nested in the ice bath. Mix in the brown sugar, add the cinnamon stick back in, then stir until cool, then chill thoroughly, preferably overnight.
Whisk in the vanilla, liquor (if using), and pumpkin. Press the mixture through a fine-mesh strainer, then freeze in your ice cream maker according to the manufacturer’s instructions.