Preheat and prepare your tin. Set your oven to 175°C (350°F). Grease a 20cm round cake tin and line the base with a circle of parchment paper. The paper should sit flat and smooth against the bottom, with no wrinkles or air bubbles.

Mash the bananas thoroughly. In a large mixing bowl, mash the bananas with a fork until they form a smooth, almost liquid purée with only a few small lumps remaining. You should see no large chunks, and the mixture will look slightly glossy and smell intensely fruity.

Combine the wet ingredients. Add the melted butter, caster sugar, eggs, and vanilla extract to the mashed bananas. Whisk vigorously for about one minute until the mixture is pale, smooth, and slightly thickened. It should look like a thin pancake batter and feel silky when you lift the whisk.

Fold in the dry ingredients. Sift the plain flour, bicarbonate of soda, baking powder, and salt directly over the wet mixture. Using a rubber spatula, fold gently until just combined — you want to see no streaks of white flour, but a few small lumps are perfectly fine. The batter will be thick and slightly lumpy, with a soft, dropping consistency.

Add the milk and mix gently. Pour in the room-temperature milk and fold it through the batter with just three or four strokes. The batter should loosen slightly and become smooth and shiny, resembling a thick milkshake. Take care not to overmix, or your cake will turn out tough.

Bake until golden and springy. Pour the batter into the prepared tin and smooth the top with the back of a spoon. Bake for 33-38 minutes, rotating the tin halfway through. The cake is ready when the top is a deep golden brown, the edges have pulled away slightly from the tin, and a skewer inserted into the centre comes out clean or with a few moist crumbs attached. The kitchen will smell intensely of toasted banana and butter.

Cool completely before frosting. Let the cake rest in the tin for 10 minutes, then turn it out onto a wire cooling rack and peel off the parchment paper. Allow it to cool completely — this takes about one hour. The cake should feel firm but springy to the touch, and the top will have a beautiful domed shape.

Make the brown butter frosting. While the cake cools, place the 100g butter for the frosting in a small saucepan over medium heat. Cook, stirring occasionally, until the butter melts, foams, and then turns a deep amber colour with a nutty, toasty aroma. You'll see brown specks forming at the bottom of the pan — this takes about 5 minutes. Immediately pour the brown butter into a heatproof bowl and let it cool for 10 minutes. Once cooled but still liquid, add the sifted icing sugar, milk, and vanilla, and beat with a hand mixer for 2-3 minutes until light, fluffy, and spreadable. The frosting will be pale beige with a rich, caramel-like fragrance.

Frost and serve. Once the cake is completely cool, spread the brown butter frosting evenly over the top and sides using a palette knife or the back of a spoon. The frosting should sit thickly on the cake, creating a smooth, creamy layer that contrasts beautifully with the tender crumb beneath.