Preheat the oven to 350º. In a medium bowl, whisk together the flour, cardamom, salt and baking soda to incorporate, then set aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream the butter with the sugars until light and fluffy, 2 minutes. With the motor running, add the eggs one at a time and beat until incorporated, scraping the sides of the bowl as needed. Add the rose water and mix to incorporate, followed by the pistachios and rose petals.
Add the flour mixture and mix until a smooth dough forms. Transfer to a parchment-lined 9-by-13-inch baking dish and, using an offset spatula, spread evenly throughout the pan.
Bake until golden brown, 15 to 20 minutes. Let the tray cool completely before slicing and serving.