Preheat the oven to 350°F (175°C). In a large bowl, whisk together the flour, sugar, baking powder, and salt. In another bowl, mix the eggs, olive oil, limoncello, lemon zest, and vanilla extract until combined. Gradually add the wet ingredients to the dry ingredients, mixing until just combined. Fold in the poppy seeds.

Bake the cake

Pour the batter into a greased 9-inch round cake pan. Bake for 35-40 minutes or until a toothpick inserted in the center comes out clean. Let it cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.

Blueberry Compote Preparation

Make the compote

In a saucepan over medium heat, combine the blueberries, sugar, and lemon juice. Cook, stirring gently, until the blueberries burst and the mixture thickens, about 10-15 minutes. Remove from heat and let cool.

Buttermilk Sorbet Preparation

Prepare the sorbet mixture

In a bowl, whisk together the buttermilk, sugar, lemon zest, and lemon juice until the sugar is dissolved. Chill in the refrigerator for about 30 minutes.

Freeze the sorbet

Pour the mixture into an ice cream maker and churn according to the manufacturer's instructions. Transfer to a container and freeze for at least 4 hours before serving.