Whisk flour, baking soda, salt, ginger, cinnamon, cloves, and pepper in a medium bowl.
Cream softened butter, brown sugar, and granulated sugar until light and fluffy, about 2 minutes.
Beat in molasses, egg, and vanilla until smooth and glossy.
Add dry ingredients in two additions, mixing just until a soft dough forms.
Divide dough in half, flatten into disks, wrap, and chill until firm, about 1 hour.
Preheat oven to 350°F. Line two baking sheets with parchment.
Lightly flour the surface. Roll one disk to 1/4 inch thickness.
Cut gingerbread men, transferring shapes to prepared sheets 1 inch apart. Re-roll scraps as needed.
Bake until set at the edges but still soft in centers, 8–10 minutes. Rotate pans halfway.
Cool cookies on the sheet 5 minutes, then move to a rack to cool completely.
For icing, whisk powdered sugar, milk, and vanilla until smooth. Adjust with drops of milk as needed.
Pipe or drizzle icing on cooled cookies. Let set before storing.