Put the pumpkin puree in a large heatproof bowl and set aside. In a medium heatproof bowl, whisk the yolks just to break them up, then whisk in the cinnamon, ginger, and half of the brown sugar. Set aside. In a heavy nonreactive saucepan, stir together the cream, milk, salt and the remaining sugar and put the pan over medium-high heat. when the mixture approaches a bare simmer, reduce the heat to medium.
Carefully scoop out about 1/2 cup of the hot cream mixture and, whisking the eggs constantly, add the cream to the bowl with the egg yolks. Repeat, adding another 1/2 cup of the hot cream to the bowl with the yolks. Using a heatproof rubber spatula, stir the cream in the saucepan as you slowly pour the egg-and-cream mixture from the bowl into the pan.
Cook the mixture carefully over medium heat, stirring constantly, until it is thickened, coats the back of a spatula, and holds a clear path when you run your finger across the spatula, 2 to 3 minutes longer. Strain the base through a fine-mesh strainer into the container with the pumpkin puree in it. Gently stir to combine. Set the container into an ice-water bath, wash your spatula, and use it to stir the base occasionally until it is cool. Remove the container from the ice-water bath, cover with plastic wrap and refrigerate the base for at least 2 hours or overnight.