In a small saucepan, whisk together the sugar and lemon zest until the zest releases its oils. You'll know it's done when you can really smell the lemons and the sugar has gone a very pale yellow.
Pour the egg yolks in (make sure you save the whites for the cake!) and whisk for about 2 minutes, until pale and thick.
Place the saucepan over low speed and pour in the lemon juice and salt. Whisk constantly until thick and glossy - this can take up to 10 minutes. Make sure you don't stop whisking! Coat the back of a spoon with the lemon curd, it should coat it very thickly - that's how you know it's done.
Turn the heat off and add the butter a piece at a time, whisking after each addition. Make sure the butter has melted before adding the next piece! Then add the vanilla extract and whisk to combine once more.
Pour the lemon curd into a jar or bowl and let cool to room temperature. Note, if you want you can strain it at this point to remove the pieces of zest. Cover the top of the lemon curd with clingfilm to stop a skin from forming. Once cooled, place in the fridge for up to 1 week but at least 2 hours before using.
Make Cake
Preheat the oven to 350°F/175°C. Grease and line two 8-inch cake pans
In the bowl of a stand mixer with the paddle attachment, or in a large bowl with a hand mixer, mix all of the dry ingredients on slow speed until blended. Add in your butter and mix on low until combined. It should look like damp sand.
Add your egg whites one at a time, and then your whole egg. Beat well after each addition, making sure you scrape the sides and bottom of the bowl so all of the cake batter is getting mixed.
In a small jug or bowl, whisk together the vanilla extract, yogurt, lemon juice and milk until combined.
Slowly pour in one-third of the milk mixture into the mixing bowl. With the speed on low, mix together. Pour in the rest of the milk in two more additions, mixing slowly after each addition. Careful as it is a liquidy batter and may spill!
Beat for 2 minutes. Your batter will be liquidy which is normal!
Pour the batter into your two pans. Bake for 25-30 minutes. A toothpick inserted into the cake will come out clean. Remove from the oven and let cool in the pans for 15 minutes. Remove the cakes from the pans and let cool on a wire cooling rack until completely cool.
Make Frosting
In a stand mixer fitted with the whisk attachment, or in a large bowl with a handheld mixer, beat the heavy cream, sugar, salt and vanilla extract on low speed for 2 minutes. The sugar should have completely dissolved.
Increase the speed to medium-high for 2 minutes and beat until it becomes very thick and looks like greek yogurt.
Change the speed to low and add the cold cream cheese, a spoonful at a time. Once it has all been added, increase the speed to medium-high and beat for another 2 minutes.
Assemble
Place one cake layer down. Use a small offset spatula to spread a thin layer of cream cheese frosting. Place some of the frosting in a piping bag and pipe around the border of the cake. Spoon out the lemon curd (I used about ½-¾ of the lemon curd!) in the middle and spread it up until the frosting border.
Place the second layer on top and press down gently. Evenly cover the sides and top of the cake with the rest of the frosting. Place in the fridge for 30 minutes to set.
Notes
My recipes have been developed and tested in metric grams. I highly recommend using a weighing scale and measuring out your ingredients the metric way. Just click on 'metric' above ingredients. I make sure that all of my ingredients are at room temperature before I start the recipe, so pull out all of the ingredients before you start baking - except for the frosting ingredients which need to be cold!Butter: I use European style butter in my recipes. This will make your frosting much creamier and richer.Salt: If you don’t have kosher salt, halve the amount of salt called for in the recipe. 1 teaspoon kosher salt = ½ teaspoon fine salt. I am using Diamond Crystal kosher salt and not Morton Salt which is much too salty! Vanilla Extract: you can use vanilla extract in the cake, or you can use Fiori di Sicilia which is what I used. This is like a lemony vanilla and is very fragrant. Lemon Curd: feel free to use store bought lemon curd instead! Storage: Keep tightly wrapped in the fridge for up to 5 days. Let it come to room temperature before eating.
Making Ahead: You can make the cake layers ahead of time. Bake them and then once completely cooled, wrap in plastic wrap tightly. Store in the fridge for up to 2 days or in the freezer for up to 2 months. If frozen, allow to defrost and come to room temperature before frosting the cake. You can make the lemon curd a week in advance, keep in a jar with a lid in the fridge.