In a bowl, whisk the pumpkin puree and vanilla extract together. Cover and refrigerate for at least 3 hours or overnight.
In another bowl, whisk together the egg yolks, cinnamon, ginger, salt, nutmeg, 1/2 cup of the cream, and 1/2 cup of the sugar. Whisk until the mixture is smooth and the sugar is mostly dissolved. Set aside.
In a heavy medium saucepan, combine the remaining 1 1/2 cups of the heavy cream and 1/2 cup of the brown sugar. Over medium heat, cook the mixture for about 5 minutes or until bubbles form around the edge of the pan. Remove from heat.
Gradually add 1/2 cup of the hot cream mixture to the egg mixture, whisking constantly. Then pour the egg mixture into the saucepan with the remaining hot cream mixture. Cook over medium heat, stirring constantly with a wooden spoon, keeping it to a low simmer (do not let it boil). Cook for 4-6 minutes or until the mixture has thickened enough to coat the back of the wooden spoon and running the tip of the spoon through the mixtures leaves a clear trail. Strain the mixture through a fine mesh sieve into a bowl.
Place the bowl in a larger bowl with water and ice cubes and stir the mixture occasionally until it has cooled down. Whisk in the chilled pumpkin puree mixture. Cover with plastic wrap, pressing it against the custard to prevent a skin from developing. Refrigerate for at least 3 hours or overnight.
Pour the custard into an , add the graham crackers and freeze according to manufacturer's directions.
Serve immediately for softer ice cream, or transfer to a freezer-safe container and freeze at least 3 hours or up to 3 days.Makes about a quart.