Preheat your oven to 350°F (175°C).
In a large mixing bowl, combine the pumpkin purée, granulated sugar, vanilla extract, and eggs. Mix until well blended.
Add the ground cinnamon, nutmeg, ginger, cloves, and salt to the pumpkin mixture. Stir until the spices are evenly incorporated.
In a separate bowl, whisk together the flour and baking powder. Gradually add this to the pumpkin mixture, alternating with the milk, and mix until smooth.
Pour the melted butter into the bottom of a greased 9x13 inch baking dish.
Carefully pour the pumpkin batter over the melted butter without stirring.
In a small bowl, mix the brown sugar and water until combined. Slowly drizzle this mixture over the top of the pumpkin batter. Do not stir; the liquid will sink to the bottom during baking.
Place the baking dish in the preheated oven and bake for 40-45 minutes, or until the cobbler is set and the top is golden brown.
Once done, remove from the oven and let it cool slightly before serving.