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In a medium saucepan, heat the butter until melted. Continue cooking at a boil, stirring often, until the milk solids turn brown and the butter starts to smell nutty. This will take 5-10 minutes, and happens quickly near the end so watch carefully to ensure the butter doesn’t burn.
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Remove fro, he heat immediately and pour into a bowl (including all the brown flecks). Stir in the vanilla (it will bubble up), then place in the freezer until it is about half solid (about half an hour).
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Preheat oven to 175° and grease and line 3 x 20cm round cake tins (if you only have one, you can use the same tin and put the remaining batter in the fridge while you wait for each layer to cook).
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Cream the chilled brown butter and sugar until pale and fluffy. Beat in the eggs one at a time, ensuring each is incorporated fully.
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In a separate bowl, sift together the flour and baking powder.
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Fold a third of the flour mixture into the creamed mixture until just combined. Fold in half the yoghurt, then remaining flour, and finally the rest of the yoghurt.
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Divide batter evenly between the three tins and flatten the tops. Bake about 20minutes, or until a skewer comes out clean and the top springs back to touch. Be careful not to overbake.
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Cool in tins.