Whisk the flour, ginger, cinnamon, cloves, allspice, baking soda, and salt together in a bowl. Set aside.
In a large bowl, beat the softened butter and brown sugar together until fluffy, about 2 minutes. Add the molasses, egg, and vanilla, and mix until smooth.
Add the dry ingredients to the wet mixture and stir just until combined. If the dough feels too soft or sticky to roll into balls, chill it for 20-30 minutes.
Scoop the dough into 1-tablespoon balls and roll each one in the granulated sugar until fully coated. Place on a parchment-lined baking sheet, spaced about 2 inches apart.
Bake at 350F for 9-11 minutes, until the edges are just set but the centers still look slightly soft.
Wait about 2-3 minutes after the cookies come out of the oven, then press a gentle indent into the center of each one with your thumb or the back of a small spoon.
Let the cookies cool fully on the baking sheet or a wire rack before filling.
Beat the cream cheese, butter, vanilla, and powdered sugar together until smooth, adding milk a teaspoon at a time until soft enough to pipe. Pipe a small amount into each thumbprint indent.