Preheat the oven to 160 degree C and put a rack in the centre.
Butter and flour a 9 inch x 9 inch square pan.
Place the butter in a skillet over medium heat. Brown the butter until the milk solids are a dark chocolate brown and the butter smells nutty.
Pour into a shallow bowl and chill in the freezer until just congealed, 15-30 minutes.
Whisk together cake flour, baking powder, and salt.
Beat the brown butter, light brown sugar, and granulated sugar in an electric mixer until light and fluffy, about 2 minutes.
Beat in the eggs one at a time, mixing well, and then the vanilla extract.
Stir in the flour mixture at low speed until just combined.
Scrape the batter into the greased and floured pan.
Smooth the top with a rubber spatula and rap the pan on the counter.
Bake until golden brown on top and when a toothpick inserted into the centre comes out clean, about 25 minutes.
Cool in the pan 10 minutes. Run a knife along the edge and invert right-side-up onto a cooling rack to cool completely.
For Strawberry Ice Cream:
Beat the egg yolks, sugar and corn flour until thick and creamy.
The sugar has to completely dissolve in the mixture.
Combine the milk and cream together and heat it just under boiling point.
At this stage immediately pour the milk into the egg mixture little by little whisking continuously.
Once it's well combined, return the mixture to the heat again and heat it in medium to low heat, whisking continuously until the mixture coats the back of the spoon.
Remove it from heat and let it cool down completely. Then add the pureed strawberries and stir well.
Churn in the ice cream machine according to your machine instructions.
Allow it to set in the freezer until required.
For Baked Alaska:
Cut a circle out of the cake using a tea cup. Line the inside of the tea cup with a cling film, add a piece of cake at the bottom, fill with ice cream and then top with another piece. Freeze it completely.
Take the ice cream after it's completely frozen, pull the cling film out and remove it from the cling film and flip it upside down.
Meanwhile, make the meringue layer.
Whisk the egg whites until soft peaks forms, add the sugar little by little and keep whisking until stiff peaks form.
Spread the meringue layer on top of ice cream layer using a spatula or using a piping bag.
Bake it in the preheated oven at maximum temperature until it's slightly brown on top.