Heat the oven to 350°F. Rub the bottom and sides of a 9x13-inch baking pan with 1 tablespoon of the unsalted butter until the pan is thickly coated.

Heat 2 cups water in a small saucepan over medium heat until steaming and just starting to simmer, 4 to 5 minutes. Turn off the heat.

Whisk 2 cups all-purpose flour, 1 tablespoon baking powder, and 1 tablespoon of the pumpkin pie spice in a medium bowl until combined.

Melt the remaining 4 tablespoons unsalted butter in a large heatproof bowl in the microwave, 45 seconds to 1 minute. Whisk in 1 cup pumpkin puree, ⅔ cup of the packed dark brown sugar, ½ cup whole milk, and 1 teaspoon of the kosher salt until smooth.

Stir the flour mixture into the pumpkin mixture until there are no pockets of dry flour; the batter will be thick.

Spoon the batter into the pan and evenly spread with an offset spatula all the way to the edges of the pan.

Mix the remaining 1 cup packed dark brown sugar, ½ cup coarsely chopped pecans, 2 tablespoons granulated sugar, remaining ¼ teaspoon pumpkin pie spice, and 1 pinch kosher salt together in a small bowl. Sprinkle over the batter in an even layer.

Carefully and slowly pour the hot water all over the cake without stirring.

Bake until the cake is well risen and the top of the cake is dry to the touch, 25 to 35 minutes. Let cool for about 15 minutes so the sauce can thicken slightly.

Use a large serving spoon to scoop the warm cake and some sauce into bowls, and top with whipped cream or vanilla ice cream if desired.