Set a rack in the center of the oven and preheat to 375°F. Grease a 9x13-inch baking pan and line it with parchment paper, letting an inch or two of excess paper hang over two opposite sides of the pan.
In a large bowl, whisk the butter, pumpkin, brown sugar, ½ cup sugar, ½ cup water, eggs, and 2 teaspoons of the vanilla. In a small bowl, whisk together the flour, cinnamon, baking powder, baking soda, 1 teaspoon kosher salt, the allspice, nutmeg, ginger, and cloves. Pour the flour mixture into the butter mixture and stir until just combined.
Pour the batter into the prepared baking pan and smooth the top with an offset spatula or the back of a spoon. Transfer to the oven and bake until a cake tester inserted in the center of the pan comes out clean, 23–25 minutes. Remove from the oven and let the cake cool completely in the pan.
When ready to serve, make the meringue: in a stand mixer fitted with the whisk attachment, add the egg whites, cream of tartar, and the remaining ¼ teaspoon salt. Whisk on medium until the eggs are frothy, 45–60 seconds, then slowly add the granulated sugar with the mixer running. Increase the mixer to high speed and continue mixing until the meringue is shiny and glossy and holds very soft peaks, about 2 minutes. Add the remaining ½ teaspoon of vanilla, and continue mixing on high speed until the meringue holds stiff peaks, about 1 minute more. (The meringue should be glossy and hold its shape when you invert the whisk, and it should feel smooth when you rub some between your fingers.) Use a rubber spatula or large spoon to mound and spread the meringue on top of the pumpkin cake, making peaks and valleys as you go. Use a kitchen torch to brown the topping lightly in places, moving the flame quickly and in small circles to avoid burning (don’t worry if an area catches flame—just blow it out). Alternatively, you can place the meringue-covered cake under a hot broiler, no closer than 8 inches to the heat source, watching closely until the tips are lightly browned.
Holding on to the excess parchment paper, carefully lift the bars out of the pan and onto a cutting board. Cut into 15 squares (about 3x3-inches in size), cleaning the knife with a damp cloth between each cut if needed. Serve immediately.