Preheat the oven to 175°C. Melt the butter gently in a small saucepan, then cool a little. Add the sugar, eggs, milk and vanillin. Sift the dry ingredients into a wide bowl and pour in the butter mixture. Stir everything together, adding the mashed bananas; do not overmix or beat the batter.
Turn the mixture into a lightly buttered 6-cup ring tin and bake for 30 minutes. The cake should be browned, slightly shrinking from the sides of the tin and a skewer inserted in the middle should come out clean.
To make the lemon icing, put the icing sugar in a bowl and add enough lemon juice to make it a barely pourable consistency. Spread over the cooled cake and leave to set.