Cake : Preheat oven to 160ºC fan bake / 180ºC bake. Grease four round cake tins (20cm, 17cm, 15cm and 12cm diameter) and line with baking paper.
In the large bowl of a stand mixer, beat caster sugar and eggs together until thick and creamy (5 minutes). Gradually beat in oil, lemon zest, lemon juice and vanilla extract.
Sift flour, baking soda and salt over the top. Mix until just combined. Stir in yoghurt, then swirl jam through the batter.
Divide mixture between prepared tins (try to fill them all to approximately the same height). Cut any large strawberry pieces in half and scatter strawberries over the cakes, gently pressing in.
Bake for 25-35 minutes, until the tops spring back when lightly pressed and a skewer inserted into the centre comes out clean (the smaller cakes will likely be ready a few minutes before the larger ones).
Place tins on wire cooling racks and allow cakes to cool completely.
Icing : In a large bowl, beat half of the butter until pale and creamy. Add half of the icing sugar, milk and strawberry essence. Beat on low to combine, then increase speed to high and beat until light and fluffy.
Place largest cake layer on a cake board or serving plate (at least 25cm diameter), and set this on a decorating turntable, if you have one. Spread cake with a layer of icing, then place the second largest cake layer on top and spread this with icing.
Use a round cookie cutter to cut a hole in the centre of each of the two smaller cake layers (approximately 4.5cm diameter, or large enough to snugly fit the lower half of the doll in). Assemble these layers on top of the two bottom layers, adding a layer of icing in between.
Apply a thin layer of icing around the outside of the whole cake. Refrigerate cake while you prepare the second batch of icing.
In a large bowl, beat remaining butter until pale and creamy. Add remaining icing sugar, milk and strawberry essence. Beat on low to combine, then increase speed to high and beat until light and fluffy. Beat in any icing leftover from the first batch.
Beat a small quantity of pink food colouring into the icing to create a very pale pink colour (noting that the colour tends to darken over time). Remove slightly less than one third of this icing into a separate bowl.
Beat a little more pink food colouring into the larger portion of icing to create a medium pink. Remove slightly less than half of this icing into a separate bowl.
Beat more pink food colouring into remaining icing to create a darker pink. Transfer this to a large piping bag fitted with a large open star piping tip. Pipe rounds of icing around the base of the cake (squeeze the icing on, with the piping tip perpendicular to the cake, and release). Repeat to form 2-3 more rows above (or to cover one third of the cake).
Transfer medium pink icing to a piping bag fitted with a large open star piping tip. Pipe rounds of icing above the previous layer. Repeat to form 2-3 more rows above (or to cover one third of the cake).
Transfer light pink icing to a piping bag fitted with a large open star piping tip. Pipe rounds of icing above the previous layer. Repeat to form 2-3 more rows above, or until the whole cake is covered.
Decorate : Wrap the lower half of the doll in clingfilm and insert it into the top of the cake. You may need to pipe a little extra light pink icing around the doll's waist. Decorate the cake with large round sprinkles / sugar pearls, if using.
Store in an airtight container in a cool room until ready to serve.
NOTES : The specified tin sizes work well to create a cone shape that doesn't need to be trimmed. If you don't wish to use four different tins, you could use 1 x 20cm and 1 x 15cm tin. Bake in two batches and trim the layers to create the cone shape.
Cake layers can be made up to 2 days in advance. Cake can be assembled up to 1 day in advance. Store in an airtight container in a cool, dry room.