Make the blackberry swirl by adding the blackberries, sugar, lemon juice and water to a medium saucepan. Bring the mixture to a boil. Lower the heat and simmer for 7 to 10 minutes, smashing the blackberries with a spoon as they simmer. Push the blackberry mixture through a mesh sieve to remove the seeds and set it aside to cool.

Pre-heat the oven to 350°F. Generously grease a 10-cup Bundt cake pan with butter and dust the pan with flour, tapping out any excess.

Combine the flour, baking powder, baking soda, salt, lemon zest and poppy seeds in a bowl.

Cream the butter and sugar in a second bowl (or in the bowl of a stand mixer using the paddle attachment), beating until the butter is light in color and fluffy. Add the yogurt, whole eggs, egg yolks, vanilla extract and lemon juice and combine. Add the dry ingredients to the bowl in two or three stages, stirring only until you see no more streaks of flour.

Pour one third of the batter into the prepared Bundt pan. Pour half of the blackberry mixture over the batter. Using a thin spatula or spoon, swirl the blackberry mixture into the cake batter. Repeat with another third of the cake batter and then the remaining blackberry mixture. Swirl the blackberry purée into the batter once again and pour the remaining cake batter on top to completely cover the blackberry layer.

Bake at 350ºF for 50 to 55 minutes, until a toothpick inserted into the middle of the cake comes out clean. Let the cake cool in the pan for 10 minutes, then unmold by inverting it onto a wire rack and allow it to cool completely.

Make the lemon glaze by whisking 1 cup of the powdered sugar, lemon zest and lemon juice together until smooth. Let the glaze sit for a few minutes and then stir again. It should form thick ribbons off the spoon. If it is too thin, add the extra tablespoon of powdered sugar.

Place the cake on the cooling rack over a sheet pan. Slowly pour the glaze over the cooled cake and let it drip down the sides of the cake. Let it rest for a few minutes. If a lot of the glaze has dripped down to the sheet pan, scrape it back into the bowl and pour another layer over the cake. Let the glazed cake rest for at least 15 minutes before serving.

Garnish the glazed cake with fresh blackberries and lemon slices and serve with some whipped cream if you like.