Preheat the oven to 350 degrees F. Butter and lightly flour a 10-inch cake pan.
In a mixer fitted with a paddle attachment (or using a hand mixer), combine the melted butter, egg, and sugar. Add the molasses and water and mix. In another bowl, sift the flour, baking soda, salt, ginger, and cinnamon together. Working in batches, mixing after each addition, add the dry ingredients to the molasses mixture. Mix until smooth.
Pour into the prepared pan and bake in the center of the oven for about 35 minutes, until browned and the top of the cake feels dry and firm to the touch. Let cool in the pan for 10 minutes, then turn out onto a wire rack to completely cool.
For the topping: In a large skillet, melt 3 tablespoons of the butter over medium heat. Add the apple slices and cook, stirring, until tender, about 10 minutes. Add 3 tablespoons of the brown sugar and stir to combine. Cook a few more seconds to dissolve the brown sugar then pour it over the cake and serve.