Preheat the oven to 350°F. Lightly butter an 8-inch square baking dish and place it on a rimmed baking sheet.
Whisk together the flour, baking powder, salt, granulated sugar, cinnamon, nutmeg, ginger, and cloves in a large bowl.
In a separate bowl, stir together the applesauce, milk, melted butter, vanilla, and grated carrots.
Add the wet ingredients to the dry ingredients and stir just until a thick batter forms. Spread the batter evenly in the prepared baking dish.
Combine the topping sugars and chopped nuts, then sprinkle them evenly over the batter.
Slowly pour the very hot water over the entire surface. Do not stir.
Bake for 40 to 50 minutes, or until the top is golden and set while the edges are bubbling.
Rest the cobbler for 10 minutes before serving warm with vanilla ice cream or whipped cream.