Preheat the oven to 325 degrees F.
Butter the sides of an 8-inch springform pan, and line the base with greaseproof or parchment paper.
In a large bowl or in an electric food mixer, cream the butter until soft. Add the sugar, and beat until the mixture is light and fluffy. Gradually add the beaten eggs, beating well between each addition.
Sift in the flour and baking powder, add the poppy seeds, and stir until combined.
Spoon the mixture into the prepared cake pan, making a slight hollow in the center with the back of a spoon. Bake in the oven until the cake is cooked in the center and a skewer inserted into the middle comes out clean, about 25 to 28 minutes.
Allow the cake to stand for 5 minutes before carefully removing it from the pan and transferring it to a wire rack to cool completely.
If you wish, when the cake is cool, split it in half using a long-bladed serrated knife. Place one layer of the cake on a serving plate and spread over 2 generous tablespoons of the vanilla buttercream, then place the other half of the cake on top. Spread the remaining icing over the top and sides with a palette knife or table knife. The cake can also be kept whole, with the icing spread over the top and sides.
Place the milk and sugar in a small saucepan and bring to a gentle boil, stirring to dissolve the sugar.
Whisk the egg yolks in a small bowl, then pour the milk onto the yolks, whisking continuously. Return the mixture to the pan and cook over a low-medium heat, stirring all the time with a wooden or silicone spatula until it thickens and the mixture just coats the back of a spoon-this may take approximately 10 minutes. Do not allow the mixture to get too hot or it will scramble (if it does, quickly remove the pan from the heat and pour the mixture through a sieve). Pour into a bowl or jug and allow to cool slightly.
Meanwhile, place the butter in a bowl (I usually do this in an electric food mixer with a whisk) and whisk until soft and light.
Gradually add the almost cooled (not cold but room temperature is good) custard to the butter, whisking all the time, until is combined. Add the vanilla extract and mix.