Preheat oven at 180°C.
Add the pistachios and cardamom seeds, and process until the nuts are finely ground—then transfer to a bowl. Add the almond meal, semolina, baking powder and salt. Mix together and set aside.
Place the butter and sugar in the bowl of an electric mixer with the paddle attachment. Beat on medium-high speed until fully combined. With the machine still running, slowly add the eggs, making sure that each batch is fully incorporated before adding the next.
Remove the bowl from the machine and add the dry ingredients, folding them in by hand and, again, taking care not to overmix. Next fold in the lemon zest, lemon juice, rose water and vanilla extract and scrape the batter into the pan. Level with an offset spatula and bake for 55–60 minutes, or until a skewer inserted into the centre of the cake comes out clean but oily.
To make the rose syrup, place all the ingredients for the syrup in a small saucepan over medium heat. Bring to a boil, stirring so that the sugar dissolves, then remove from the heat. Don’t worry that the consistency is thinner than you might expect; this is how it should be.
As soon as the cake comes out of the oven, drizzle all of the syrup over the top. It is a lot of syrup, but don’t lose your nerve—the cake can take it! Sprinkle with the finely chopped pistachios and set the cake aside in its pan to come to room temperature. Remove from the pan and scatter the rose petals over the cake. Serve immediately.