Bring water, corn syrup, cinnamon, cloves, and ginger to a boil in a saucepan. Pour over butter and sugar and beat until blended. Gradually stir in flour and baking soda until dough is smooth.
Halve the dough and form into discs. Wrap discs in plastic wrap and refrigerate, 8 hours to overnight.
Preheat the oven to 375 degrees F (190 degrees C).
Let dough soften slightly. Roll it out to 1/8-inch thickness. Cut dough using gingerbread men cookie cutters. Arrange on ungreased cookie sheets.
Bake in the preheated oven until tops look dry, 8 to 10 minutes. Cool on wire racks.
Beat powdered sugar, butter, and vanilla extract together until stiff. Thin mixture out with milk. Spread frosting over cooled cookies.
