Preheat the oven to 350°F (175°C). Grease a 9x13-inch baking dish with butter or nonstick spray.
Arrange the sweet potato slices in an even layer at the bottom of the prepared baking dish.
In a medium bowl, cream together the granulated sugar and softened butter until light and fluffy. Add the flour, baking powder, and salt, mixing until combined. Gradually stir in the milk and vanilla extract to form a smooth batter.
Pour the batter evenly over the sweet potatoes, spreading it gently with a spatula to cover the slices completely.
Sprinkle the brown sugar evenly over the batter, then carefully pour the boiling water over the top. Do not stir.
Bake in the preheated oven for 50-60 minutes, or until the top is golden brown and bubbling around the edges. The sweet potatoes should be tender when pierced with a fork.
Remove from the oven and let the cobbler cool slightly before serving. The sauce will thicken as it cools.