Make the Lemon Curd

1. Whisk together the egg, granulated sugar, cornstarch (if using), lemon juice, and lemon zest until smooth.
2. Cook over low to medium heat, whisking constantly, until the mixture thickens and coats the back of a spoon.
3. Remove from the heat and whisk in the softened butter until fully melted and smooth.
4. Strain through a fine-mesh sieve, then transfer to a bowl and press plastic wrap directly onto the surface.
5. Refrigerate until completely chilled.

Make the lemon buttercream

1. Place the very soft butter in the bowl of a stand mixer fitted with the whisk attachment. Whip on high speed for 5 to 7 minutes, until very pale and fluffy.
2. Add the sifted powdered sugar in two additions. Mix slowly at first, then increase the speed and beat until fully incorporated.
3. Add the vanilla extract and heavy cream. Whip again until light, smooth, and creamy.
4. Add the cooled lemon curd and beat for about 2 minutes, until the frosting is silky and evenly combined.
5. Finish mixing on low speed to smooth the texture and remove air bubbles.
6. Use immediately, or cover with plastic wrap pressed directly onto the surface and refrigerate until ready to use.