Set the oven to 350°F with a rack at the center, then butter a 9-inch springform pan, line it with parchment, and lightly butter the paper. Cover the bottom and sides with a thin coating to help the delicate cake release.
Whisk both flours with the ground cardamom, baking powder, baking soda, and salt in a medium bowl. Break up visible clumps so the leaveners and spice spread evenly through the mixture.
Work the lemon zest into the sugar with your fingertips in the mixer bowl, then add the softened butter and beat with the paddle on medium-high for about 5 minutes, scraping the bowl as needed. The sugar should smell citrusy before mixing, and the creamed mixture should become pale and fluffy; a handheld mixer also works.
Beat in each egg separately on medium-high, incorporate the yogurt until smooth, stir in the rose water, and gently fold in the dry mixture with a flexible spatula. Stop once the flour disappears, leaving a very thick batter rather than a flowing one.
Scrape the batter into the prepared pan, level it with an offset spatula, bake for about 35 minutes, and place the pan on a rack over a baking sheet. Look for a golden surface and a clean toothpick from the center before taking the cake out.
Begin the syrup as the cake starts cooling by stirring the sugar and lemon juice in a small saucepan over medium heat for about 1 minute, then stir in the rose water and take the pan off the heat. The sugar should dissolve before the rose water goes in.
Pierce the warm cake across the top with a toothpick, small skewer, or fork, spoon the syrup evenly over it, and leave it in the pan for 30 minutes. This pause lets the syrup soak in while the fragile almond crumb begins to firm.
Release the cake from the springform pan and leave it on the cooling rack until completely cool. The surface must no longer feel warm before you move on to icing.
Whisk the powdered sugar with 1 tablespoon lemon juice and ½ tablespoon rose water, adding more liquid only as needed within the listed ranges. A smooth, opaque icing that pours slowly is the goal, rather than a thin glaze that immediately runs away.
Drizzle the icing across the cooled cake and scatter over the optional slivered or chopped pistachios and dried rose petals. Let the thick icing form visible drips while leaving some of the cake's edges exposed if desired.