Preheat your oven to 160°C, (140°C fan, gas 3). Grease and line three 20cm round cake tins. Dust with flour and knock out the excess flour.
Whisk together the flour, baking powder, baking soda, and salt in a large bowl; set aside.
In a large bowl, beat the butter on medium speed, using an electric whisk, until creamy, for around 3 to 4 minutes. Add the sugar and vanilla extract, and beat on medium speed until pale and fluffy, for about 3 minutes. Scrape down the sides of the bowl, add the egg, and beat until just combined.
In a small bowl, whisk together the milk, and the yoghurt.
With the mixer on low speed, add the flour mixture in three additions, alternating with the water mixture, beginning, and ending with the flour mixture. Scrape down the sides of the bowl. With the mixer on medium speed, beat for a few seconds to incorporate. Scatter the sprinkles across the top of the batter and fold them in with a rubber spatula.
In a medium bowl, whisk the egg whites and lemon juice just until soft peaks form, do not overbeat. Gently fold egg whites into the batter with a rubber spatula.
Divide the batter evenly among the prepared pans and smooth the tops. Bake, rotating the pans halfway through the baking time, until a toothpick inserted in the center of each cake comes out clean, 40 to 45 minutes. Transfer the cakes to a wire rack and let cool for 20 minutes. Invert the cakes onto the rack, remove from the pans, and let cool completely. Remove the baking parchment.
Combine the egg whites, sugar, and salt in the heatproof bowl over a pan of simmering water. Whisk constantly until the mixture is warm to the touch and the sugar has dissolved (the mixture should feel completely smooth when rubbed between your fingertips).
Using an electric whisk, starting on low and gradually increasing to medium-high speed, whisk until stiff peaks form. Continue mixing until the mixture is fluffy and glossy, and completely cool (test by touching the bottom of the bowl), about 10- 15 minutes.
With the electric whisk on medium-low speed, add the butter a few tablespoons at a time, mixing well after each addition. Once all butter has been added, whisk in the vanilla extract. Continue beating on low speed until all air bubbles are eliminated, about 2 minutes. Scrape down the sides of the bowl with a flexible spatula, and continue beating until the frosting is completely smooth. Keep buttercream at room temperature if using the same day.
Place one cooled cake layer on a cake board. Using a serrated knife, trim the top of the cake to create a flat surface, and evenly spread about 1 cup of buttercream over the top. Add the next layer, trim it, and frost it as before, then add the third layer (trim the final layer for a completely flat top, or feel free to leave it domed for an old-school appearance).
Frost the sides and top with the remaining buttercream. Grab a fistful of sprinkles and toss the sprinkles at the sides of the cake. Continue throwing the sprinkles until the cake (including the top) is covered in sprinkles. Chill the cake for about 5 minutes to set before serving.
Use kitchen scales to make sure each sponge cake is equal in size. You should have about 500g of cake batter per cake.
This cake will keep in a cake saver refrigerated for up to 3 days. Bring to room temperature for at least 2 hours before serving.