Preheat your oven to 350 degrees F (175 degrees C) and generously grease a 9x9-inch baking dish with butter.

In a medium saucepan, combine the maple syrup, brown sugar, boiling water, and 1/4 cup of butter for the syrup. Bring to a gentle simmer over medium heat, stirring until the sugar dissolves and the butter melts, about 3 to 4 minutes.

Remove the saucepan from the heat and carefully pour the hot maple syrup mixture directly into the prepared baking dish.

In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set this dry mixture aside.

In a large bowl, cream the softened 1/2 cup of butter and granulated sugar together using a hand mixer on medium speed until light and fluffy, about 2 minutes. Beat in the egg and vanilla extract until well combined.

Gradually add the dry flour mixture to the wet ingredients, alternating with the whole milk. Mix on low speed until just combined, being careful not to overmix the batter.

Drop spoonfuls of the thick cake batter gently and evenly over the hot syrup in the baking dish. The batter will spread and rise during baking, absorbing the rich sauce beneath it.

Bake in the preheated oven for 35 to 40 minutes, or until the cake is golden brown on top and a toothpick inserted into the center of the cake comes out clean.

Let the pudding cool for 10 to 15 minutes before serving so the sauce can thicken slightly. Serve warm, spooning the extra maple sauce from the bottom over the cake, accompanied by a scoop of vanilla ice cream.