Pumpkin Ginger Cupcakes with Ginger Cream Cheese Frosting

For the Cupcakes

finely chopped crystallized ginger

water

all-purpose flour

instant butterscotch pudding mix

baking soda

kosher salt

ground cinnamon

ground ginger

ground allspice

butter, room temperature

white sugar

light brown sugar, packed

large eggs

pure vanilla extract

pumpkin puree

For the Cream Cheese Frosting

cream cheese, softened to room temperature

unsalted butter, softened to room temperature

confectioners’ sugar, sifted

pure vanilla extract

ginger syrup from above recipe