Prepare a steamer by bringing water to a boil over medium heat.
In a large mixing bowl, whisk together the eggs and granulated sugar until the mixture becomes pale and slightly fluffy, about 3-4 minutes.
Add the mashed bananas, vegetable oil, and vanilla extract to the egg mixture. Stir until well combined.
In a separate bowl, sift together the all-purpose flour, baking powder, baking soda, and salt.
Gradually add the dry ingredients into the wet ingredients, folding gently until just combined. Do not overmix to keep the cake fluffy.
Grease a 20 cm (8 inch) round cake pan or line it with parchment paper.
Pour the batter into the prepared pan and smooth the top with a spatula.
Place the cake pan into the steamer and cover with a clean kitchen towel to prevent water droplets from falling onto the cake.
Steam the cake over medium heat for 40 minutes or until a toothpick inserted into the center comes out clean.
Carefully remove the cake from the steamer and let it cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.
Slice and serve the banana cake kukus as a delightful snack or dessert.