Preheat the oven to 350°F/180°C. Gently loosen and separate the kadaif with your hands, then spread it evenly in a wide rimmed baking sheet or roasting pan. Scatter the cold butter cubes evenly over the top. Bake for 10 to 15 minutes, stirring once or twice during baking, until the kadaif is golden brown and crisp.
While the kadaif bakes, combine the sugar and water in a small saucepan and bring to a boil over medium-high heat. Reduce the heat slightly and simmer for 2 to 3 minutes, until the sugar is fully dissolved and the syrup is clear. Set aside.
Remove the kadaif from the oven and let cool completely in the pan. Once cooled, pour the warm syrup evenly over the kadaif. Let it rest while you prepare the filling.
In a large bowl, whisk the cold heavy cream, milk, and pudding mix until thick and fluffy, about 3 to 4 minutes using a hand mixer or stand mixer on medium-high speed. Divide the mixture in half. Fold the pistachio paste into one portion until fully combined, leaving the other half plain.
To assemble, press one-third of the syrup-soaked kadaif into the bottom of a 9x13-inch/23x33 cm baking dish. Spread the vanilla cream over one half and the pistachio cream over the other half, smoothing each layer with an offset spatula. Cover with the remaining kadaif, pressing lightly to compact.
Dust the top with cocoa powder and sprinkle with chopped pistachios. Cover and refrigerate for at least 2 hours, or overnight, before slicing and serving.