In a large bowl or stand mixer, cream together butter and sugars until light and fluffy. Beat in egg and vanilla, scraping the bowl as needed. Mix in baking soda, baking powder, salt and spices. Mix in flour and cocoa powder, followed by pumpkin puree. Continue mixing until fully combined and no streaks remain.
Bake for 18-20 minutes or until a toothpick inserted comes out clean. Cool in pan until easy to handle, then transfer to a wire rack to cool completely.
Prepare frosting. In a large bowl or stand mixer, beat together butter and shortening. Mix in vanilla, cinnamon and brown sugar. Scrape bowl as needed. Mix in pumpkin puree, followed by one cup of powdered sugar at a time. Continue mixing until fully combined. Transfer frosting to a large piping bag fit with a closed star tip, or other desired piping tip.
Frost cupcakes as desired.
Store cupcakes in an airtight container or in the refrigerator for up to several days.