Preheat oven to 180°C. Grease a 20cm (base measurement) square cake pan and line the base and sides with baking paper. Melt the butter in a small saucepan over medium heat. Cook, stirring, for 2 mins or until the butter is golden brown. Set aside to cool slightly.
Pour the butter into a large bowl. Add the combined sugar, apple sauce and egg and whisk to combine.
Place the flour, bicarbonate of soda, cinnamon and salt in a large bowl. Add the apple sauce mixture and stir until just combined. Spoon the mixture into the prepared pan and smooth the surface.
Bake for 30-35 mins or until a skewer inserted in the centre of the cake comes out clean. Set aside in pan for 10 mins to cool slightly before transferring to a wire rack to cool completely.
Meanwhile, to make the burnt butter icing, melt butter in a small saucepan over medium heat. Cook, stirring, for 3 mins or until butter is golden brown. Transfer to a heatproof bowl. Place in the fridge for 1 hour or until firm.
Use an electric mixer to beat the butter in the bowl until very pale. Gradually add icing sugar, in batches, beating well after each addition. Beat for 4 mins or until pale and fluffy.
Transfer the cake to a serving plate. Spread the burnt butter icing evenly over the top. Cut into pieces and serve drizzled with caramel topping.