Combine the pumpkin puree, sugar, butter, ginger, and cinnamon in a small saucepan and heat over low heat, stirring continuously until the butter has melted and the sugar has dissolved. Turn up the heat to medium and continue to heat the mixture, stirring constantly, until the mixture comes to a boil (the mixture is quite thick, so you need to look for slow bubbles). Once the puree is bubbling, turn the heat down to medium-low and cook the mixture for about 5 minutes, still stirring. Remove the puree from the heat. For a fine consistency, use a blender (or stick blender) to blitz. Cool to room temperature, then chill in the refrigerator for at least 30 minutes before using.
In the bowl of a stand mixer fitted with a paddle, beat the butter on medium speed until smooth and creamy, 2-3 minutes. Add the salt and vanilla, and beat again until combined. Turn the mixer to low, and slowly add the powdered sugar, mixing until combined. Add the chilled pumpkin puree to the buttercream one tablespoon at a time, mixing well after each addition. Keep adding until the desired pumpkin taste is reached, making sure not to add too much puree (which can make the buttercream runny). Scrape down the sides and increase the mixer speed to medium; beat until the buttercream is light and smooth, 4-6 minutes.