Preheat the oven to 350°F. Grease a 9 by 13-inch baking pan with nonstick cooking spray.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, ginger, cinnamon, cloves, and nutmeg to combine.
In a large bowl, use an electric mixer to beat the butter and brown sugar on medium-high speed until well combined and smooth, 1 to 2 minutes. Add the egg and molasses and beat until combined. On low speed slowly add the flour mixture and beat until incorporated.
Press the dough evenly into the prepared baking pan. Bake for 15 to 20 minutes, or until cooked through but still moist and soft. Can be made ahead of time and stored in an airtight container at room temperature for up to 3 days.
Make the frosting:
In the bowl of a stand mixer fitted with the paddle attachment, beat the cream cheese, butter, and vanilla on medium-high speed until very light, creamy, and smooth. On low speed gradually add in the sugar and beat until fluffy.
Spread the frosting all over the cookie. Cut into squares and serve. Cover and refrigerate any leftovers for up to 3 days.