For the Pumpkin Purée: Split the pumpkins in half and place facedown on a sheet pan lined with parchment paper. Placing them facedown will help dry out the pumpkins, to avoid adding excessive moisture to the recipe; pumpkins are almost 94 percent water.
Bake the pumpkins until the flesh is very tender, about 20 minutes.
Peel the skin off (it should come right off if the pumpkin is properly cooked).
Purée the pumpkin in a blender until smooth, about 2 minutes.
For the Infused Milk: Bring the milk to a simmer with the spices over medium-high heat; remove from the heat and steep for 10 minutes. Strain the milk.
For the Ice Cream Base: Make the ice cream base according to the Classic Ice Cream Method. Add the pumpkin purée to the ice cream base while it is still hot. Whisk thoroughly to completely incorporate the pumpkin purée.
Strain the base through a fine-mesh strainer and cool over an ice bath. Age the base under refrigeration overnight.
Once the base has aged, churn to the desired consistency.
Harden for at least 2 hours and up to 4 hours. Reserve until needed.