Preheat oven to 350 degrees. Generously grease a 9-inch springform pan and set aside.
Whisk together the flour, pistachio pudding mix, sugar, baking powder, and salt in a medium bowl. Set aside.
In a mixing bowl, add the eggs, ricotta, butter, and vanilla. Mix until smooth.
Gradually add in the dry ingredients just until blended.
Fold in the chopped pistachios.
Spoon the cake batter into the prepared pan and spread it evenly.
Bake for 50-60 minutes, or until golden brown and cake is set in the middle.
Allow the cake the cool. Then, remove it from the springform pan.
Sprinkle powdered sugar on top, if desired, and serve.