Preheat oven to 175 degrees Celsius and grease an 8-inch cake pan.
In a large mixing bowl, whisk eggs and sugar until pale and slightly thickened.
Add the mashed bananas, milk, oil, and vanilla extract, whisking until thoroughly combined.
Sift the flour, baking powder, and salt into the wet ingredients.
Gently fold the dry ingredients into the batter until just combined, avoiding over-mixing.
Transfer the batter into the prepared pan and level the surface.
Bake for 30 to 35 minutes or until a wooden skewer inserted into the center comes out clean.
Cool the cake in the pan for 10 minutes before inverting onto a wire rack to cool completely.