For the cake:
Preheat oven to 350F. Butter and flour a 13-by 9-inch cake pan, knocking out any excess flour.
Pulse pistachios in a food processor until finely ground (do not over process into a paste). Add baking powder, cardamom, salt and pulse to combine.
Combine milk and vanilla in a measuring cup. Beat together butter and sugar in a large bowl with an electric mixer until pale and fluffy. Add eggs 1 at a time beating well after each addition. Alternately add pistachio, flour and milk in batches and mix at low speed until combined.
Spread batter evenly in cake pan and bake until tester comes out clean, about 30 minutes.
For the frosting:
Whisk all ingredients until smooth. Cover and refrigerate. Drizzle over cake slices when serving.