Preheat the oven to 350°F.
Add the butter to a 9x13-inch baking dish and place it in the oven for 5–10 minutes, or until completely melted. Remove the baking dish from the oven.
In a large bowl, whisk together the mashed sweet potato, brown sugar, milk, and vanilla extract until smooth.
Add the flour, baking powder, salt, cinnamon, ginger, and nutmeg. Stir just until the batter comes together.
Drop spoonfuls of the batter evenly over the melted butter in the baking dish. Do not stir.
Sprinkle the brown sugar evenly over the batter, followed by the pecans. Carefully pour the boiling water evenly over the entire dish. It will look unusual at this stage, but don't mix anything together.
Bake for 30–40 minutes. The top should be set and lightly golden while the buttery pecan layer remains rich and gooey underneath. For the gooey texture that makes this recipe special, avoid baking much longer than 40 minutes.
Let the cobbler cool for 10–15 minutes, then serve warm with a generous scoop of vanilla ice cream.