Prepare a steamer by bringing water to a boil and greasing a 20 cm round cake pan with vegetable oil.
In a large mixing bowl, whisk together the eggs and granulated sugar until the mixture becomes pale and slightly fluffy.
Add the mashed bananas, vegetable oil, and vanilla extract to the egg mixture. Mix well until fully 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 banana mixture, stirring gently until just combined and smooth. Avoid overmixing.
Pour the batter into the prepared cake pan and smooth the surface with a spatula.
Place the cake pan into the steamer and cover the lid with a clean kitchen towel to prevent water droplets from falling onto the cake.
Steam the cake over medium heat for 40 to 45 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 to a wire rack to cool completely.
Slice and serve the banana cake kukus as a delightful snack or dessert.