Puree will keep 3 days in the fridge or 6 months in the freezer. (Thaw before using.)
Preheat oven to 375 degrees F. Line a standard 12-cup muffin tin with cupcake liners or grease very well.
In a medium mixing bowl, combine flour, baking soda, salt, and spices.
In the bowl of a stand mixer fitted with the paddle attachment, beat butter and brown sugar until light and fluffy, about 3 minutes. Stop and scrape down sides as needed. Add eggs one at a time. With mixer on low, add squash puree and vanilla (the mixture may curdle slightly at this point, but it's fine). With mixer on low, fold dry ingredients into the wet, mixing just until incorporated.
In the bowl of a stand mixer fitted with the paddle attachment, beat cream cheese and butter until light and fluffy. Add powdered sugar in three batches, fully whipping it into the mixture before adding more. Beat in brown sugar and vanilla. Set frosting in fridge for 10 minutes.
*Note: For a more firm (and sweet) frosting, add up to 2 cups powdered sugar