Preheat the oven to 350°F. Lightly grease a 9-by-9-inch or 2 ½- or 3-quart casserole dish (no more than 4 inches deep).
For the cake, place the pistachios and walnuts on a sheet pan and bake until lightly toasted, 6 to 7 minutes. Allow to cool, then coarsely chop and set aside.
In a large bowl, stir together the flour, cinnamon, baking soda, 5-spice powder, ginger, nutmeg and salt, then set aside.
In the bowl of a stand mixer fitted with the whisk attachment, add the oil and eggs and mix until well blended and air bubbles appear. Add the brown sugar and vanilla extract and mix until combined. Add the dry ingredients in batches, mixing until just combined.
Using a spatula, fold in the carrots, orange zest and chopped nuts. Pour the batter into the prepared pan and bake until a toothpick or knife inserted in the center comes out clean, 35 to 45 minutes. Cool completely before frosting.
For the frosting, in the bowl of a stand mixer fitted with the whisk attachment, beat together the cream cheese and butter until completely smooth and fluffy. Add the confectioners' sugar in batches, followed by the whole milk and vanilla, and mix until thoroughly combined. Spread evenly over the cooled cake and garnish with pistachios.
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