Mix spice cake mix, pumpkin puree, chocolate chips, nutmeg, cinnamon, ginger, and white sugar in a large bowl until just combined. Roll dough into 1-inch balls and place 1 inch apart onto ungreased baking sheets.
Bake in the preheated oven until golden brown, 11 to 13 minutes.
Mix confectioners' sugar, milk, and vanilla extract together in a bowl until smooth, adding more milk if needed. Drizzle sugar mixture over warm cookies.