Step 1
Heat the oven to 350 degrees with the rack in the center position. Lightly butter the bottom and sides of a 9-inch springform cake pan, then line with parchment paper and very lightly butter the parchment.
Step 2
Prepare the cake: In a medium bowl, whisk to combine the almond flour, all-purpose flour, cardamom, baking powder, baking soda and salt.
Step 3
In the bowl of a stand mixer, rub the sugar and lemon zest together with your fingertips to release the perfume of the zest. Add the butter and beat with the paddle attachment on medium-high speed until light and fluffy, scraping down the sides as needed, about 5 minutes. (You may also use a handheld mixer.)
Step 4
Add the eggs one at a time, beating on medium-high until combined. Beat in the yogurt until smooth, then stir in the rose water. Fold in the dry ingredients using a flexible spatula until just combined, with no visible flecks of flour. It will be a very thick batter.
Step 5
Transfer the batter to the prepared cake pan and spread evenly, smoothing the top with an offset spatula. Bake until golden brown and a toothpick inserted in the middle comes out clean, about 35 minutes. Transfer pan to a cooling rack set over a baking sheet.
Step 6
As soon as you have left the cake to cool, prepare the syrup: In a small saucepan, combine the sugar and lemon juice over medium heat, stirring until the sugar dissolves, about 1 minute. Stir in the rose water and remove from the heat.
Step 7
While the cake is still warm, use a toothpick, a small skewer or a fork to poke holes across the surface of the cake. Using a spoon, drizzle the syrup evenly across the surface. Allow the cake to absorb the syrup and cool for 30 minutes in the pan.
Step 8
Remove the cake from the pan and let cool completely on the rack.
Step 9
When the cake has fully cooled, prepare the icing. (Do not ice the cake while warm.) In a small bowl, whisk together the powdered sugar, 1 tablespoon of the lemon juice and 1/2 tablespoon of the rose water until the icing is smooth, thick and pourable. Add more lemon juice or rose water if needed to thin the icing.
Step 10
Drizzle the icing over the cake, top with pistachios and dried rose petals, if using. (The cake will keep covered on the counter for up to 2 days and 3 days in the refrigerator.)