Heat your oven to 400 degrees F. Place the acorn squash halves cut-side up on a baking sheet. Brush the cut surfaces lightly with oil or melted butter. Roast for 35 to 40 minutes until the flesh is completely tender when pierced with a fork and the edges are starting to caramelize. Let them cool until you can handle them comfortably.

Scoop the squash flesh away from the skin and transfer it to a blender or food processor. Blend until completely smooth. You need 1/2 cup of puree for this recipe. If you have extra, save it for soup or another baked good. Reduce the oven temperature to 350 degrees F and grease an 8x8 baking dish.

In a medium bowl, whisk together the flour, the first 1/2 cup of brown sugar, baking powder, cinnamon, cloves, and salt. Add the squash puree, milk, melted butter, and vanilla. Stir with a rubber spatula until everything is just combined and a thick batter forms. Scrape it into the prepared baking dish and spread it into an even layer.

Sprinkle the remaining 3/4 cup of brown sugar evenly over the top of the batter. Do not stir it in. Now slowly pour the hot water over the entire surface. The dish will look very loose and messy at this point. That is exactly what you want. Do not stir anything.

Set the baking dish on a larger rimmed baking sheet to catch any bubbling. Bake at 350 degrees F for 33 to 37 minutes. When it is done, the top layer will look set and slightly springy, with a caky texture when tapped. The edges will be bubbling around the sides. There will be a saucy layer underneath that you can see moving slightly when you gently shake the dish.

Let the pudding cake rest for 5 minutes before serving. Spoon it into bowls, making sure to scoop down through the cakey top and get some of the warm pudding sauce from underneath. Serve with vanilla ice cream or unsweetened whipped cream if you like, though it is very good on its own.