Butter an 8- or 9-inch round baking pan and line it with parchment paper. Preheat the oven to 350° F.
In a stand mixer fitted with a paddle attachment, cream the butter and the sugar until pale and fluffy, 3 to 5 minutes.
Add the eggs one at a time, beating after each addition. Add the ground pistachios and almond meal and mix to combine. Add the lemon juice, lemon zest, and rose water and mix well.
Whisk the flour and cardamom together in a small bowl. (If you have used unsalted pistachios, add 1/2 teaspoon of salt at this stage.) Then use a rubber spatula to gently fold this dry mixture into the wet butter and egg mixture. Be careful not to overmix.
Transfer the mixture into the prepared cake pan, spread it out evenly, then bake for 40 minutes. Tent the top of the cake with aluminum foil and bake for another 10 minutes, or until the sides are starting to brown, a toothpick inserted into the center of the cake comes out clean, and the cake is no longer wet-looking or sticky. Allow the cake to cool completely in the pan for several minutes, then transfer to a wire rack to cool completely.
To make the icing, whisk together the lemon juice and confectioners' sugar. Pour this over the cooled cake, then wait for at least 30 minutes for the icing to set before slicing and serving.
A (former) student of English, a lover of raisins, a user of comma splices. My spirit animal is an eggplant. I'm probably the person who picked all of the cookie dough out of the cookie dough ice cream. For that, I'm sorry.