Heat the oven to 325°F (165°C). Grease an 8-inch square baking pan and line it with parchment paper, leaving a little overhang to help lift out the cooled cake.

Separate the eggs, placing the whites in a clean bowl and the yolks in a large mixing bowl. Beat the egg whites until stiff peaks form, then set them aside while you prepare the pumpkin batter.

Beat the egg yolks with the powdered sugar, cooled melted butter, water, and vanilla extract until smooth and well combined. Mix in the pumpkin purée and pumpkin pie spice until the color is even.

Add the flour and mix only until it is incorporated. Gradually whisk in the lukewarm whole milk; the finished batter will be thin, which is expected for this cake.

Gently fold the whipped egg whites into the batter in several additions. Stop while some white streaks remain instead of mixing until completely smooth.

Pour the batter into the prepared pan and bake for 60–70 minutes. The top should be set and lightly browned, while the center should retain a slight jiggle.

Cool the cake completely in the pan for at least 1 hour. Refrigerate it afterward if you want especially neat slices, then dust with powdered sugar before serving.