Preheat the oven to 180°C/350°F (without fan).
Generously grease a round baking dish measuring 24 cm/9.5 inches.
In a large mixing bowl, whisk together the eggs and sugar until the mixture is a pale yellow.
Add the melted butter, coconut milk, vanilla extract, and pumpkin purée.
Whisk to combine.
Slowly whisk in the glutinous rice flour, salt, cinnamon, ginger, and nutmeg.
Keep whisking until you have a thick and smooth batter. You don't have to worry about over-mixing because this batter does not contain any gluten.
Pour the batter into the greased baking dish.
Tap the baking dish on the bench top a few times to bring any air bubbles to the surface.
Bake the cake for about 40-55 minutes, or until the cake is lightly golden all over and the centre feels firm when lightly touched. The cake should puff up slightly during baking, which will indicate that it is cooked all the way through. Upon cooling, the cake will settle a little.
Place the baking dish on a wire rack and allow the cake to cool (in the baking dish) for at least 30 minutes.
Use a sharp knife to slice the cake and serve immediately.
This mochi cake tastes best on the day it is baked because the topping and the edges will be nice and crispy. Any leftovers can be kept in the fridge and reheated (see below), but it will lose its crispness.