Preheat your oven to 375°F (190°C) and lightly grease an 8-inch square baking dish with butter.

In a medium mixing bowl, whisk together the flour, baking powder, and salt until well combined.

In a separate large bowl, cream the softened butter and granulated sugar together using a hand mixer or whisk until light and fluffy, about 2-3 minutes.

Beat in the egg and vanilla extract until the mixture is smooth and pale in color.

Gradually add the dry ingredients to the butter mixture, alternating with the milk, starting and ending with the flour mixture; stir just until combined and no dry streaks remain.

Spread the thick cake batter evenly into the prepared baking dish using a spatula.

In a small saucepan, combine the maple syrup, heavy cream, and brown sugar; bring to a gentle simmer over medium heat while stirring.

Carefully pour the hot liquid mixture over the back of a spoon onto the cake batter, covering it completely (do not stir the sauce into the batter).

Bake in the preheated oven for 35 to 40 minutes, or until the cake is golden brown on top and the sauce is bubbling vigorously around the edges.

Let the pudding cool for at least 15 minutes before serving warm, allowing the sauce to thicken slightly at the bottom of the dish.