Put butter in the bowl of a stand mixer fitted with the paddle attachment and mix on low speed until butter is smooth with no lumps. With the motor running, add the sugar, 1 cup at a time, adding the next cup only after the first addition has been integrated into the mixture. Add the applesauce and cinnamon.
Stop the machine and add the vanilla and salt. Paddle on low-medium speed until completely smooth, approximately 2 minutes. Add the water and continue to mix until light and fluffy, 2-3 minutes.
The frosting can be kept in an airtight container at room temperature for up to 2 days.