Preheat the oven to 400°F (200°C). Butter a 9-in (23 cm) square baking pan.
For the sauce:
In a saucepan, whisk the maple syrup and heavy cream (or the brown sugar, water, and flour) together. Bring to a boil, whisking occasionally. Turn off the heat and pour the sauce into the prepared baking dish.
For the cake:
In a small bowl, mix the flour, baking powder, and salt together. Set aside.Measure the milk in a cup and mix in the vanilla extract. Set aside.
In the bowl of a stand mixer, or in a large mixing bowl if using a hand mixer, beat the butter and sugar together until fluffy. Mix in the eggs one at a time, beating well after adding each one. With the beater on low speed, mix in half of the dry ingredients, then mix in the milk, and finally add the remaining dry ingredients, mixing just until combined. Spoon the cake batter over the hot sauce.
Set the baking dish over a baking sheet to prevent spills. Bake the maple pudding for about 45 minutes, or until the top is golden brown and the sauce is bubbly.
SERVING: To serve the maple pudding, spoon some of the warm cake on serving plates and enjoy with vanilla gelato or ice cream.
STORAGE: Store the maple pudding covered and refrigerated for up to 5 days. Warm up servings in the microwave or in a low oven before enjoying.
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