Preheat your oven to 350°F (175°C) and grease a 9x9-inch baking dish.
In a large mixing bowl, whisk together the pumpkin puree, granulated sugar, brown sugar, milk, and vanilla extract until smooth.
Add the self-rising flour, pumpkin pie spice, and salt. Mix until well combined. Pour the batter into the prepared baking dish.
Drizzle the melted butter evenly over the batter. Sprinkle the chopped pecans and brown sugar over the top.
Carefully pour the hot water over the entire mixture, ensuring the pecans and sugar are not disturbed. Do not stir.
Bake in the preheated oven for 40-45 minutes, or until the top is golden and the cobbler is set.
Allow the cobbler to cool for 10 minutes before serving. The bottom will have a gooey pumpkin layer, and the top will be crispy and caramelized.