In a saucepan, melt the butter along with the sugar, syrup, molasses, ginger, and cinnamon. Off the heat, add the milk, eggs, and baking soda in its water.
Measure the flour out into a bowl and pour in the liquid ingredients, beating until very well mixed (it will be a very liquid batter). Pour it into the pan and bake for 3/4 - 1 hour until risen and firm.
Be careful not to overcook it, as it is nicer a little stickier, and anyway it will carry on cooking as it cools.
And when it is cool, begin the frosting. Whisk the lemon juice into the confectioners' sugar first, then gradually add extract and milk, as needed to make desired consistency.
Spread over the cooled gingerbread with a palette knife, and leave to set before cutting.