Preheat the oven to 350°F / 180°C. Grease and line a 8-inch/20 cm round cake tin.
Place the eggs in a large bowl and whisk well. Add the coconut milk, butter, brown sugar, ginger, cinnamon and salt and whisk again. Scatter in the grated carrot and glutinous rice flour and whisk for about 2 minutes, until well incorporated and the batter is smooth and free of dry flour. Gently tap the cake tin against the countertop 3 times to burst any air bubbles.
Pour the batter into the prepared tin and bake for 55–60 minutes (or more), until the top looks slightly crackly and it springs back when gently pressed.
Remove from the oven and allow to cool completely, about 60 minutes. Run a sharp knife around the edge of the cake to loosen it, then cut into slices or squares while still in the dish. If your knife has problems cutting through the sticky cake, let it cool for a while longer or lightly dust your knife with cornstarch (cornflour) – the longer it cools, the easier it will be to cut.