In the bowl of a stand mixer fitted with the paddle attachment, beat the butter, granulated sugar, and brown sugar until light and fluffy, about 2 to 3 minutes. Beat in the egg until just combined. Add the flour, baking soda, salt, pistachios, and chopped milk chocolate. Mix until a soft dough forms. Chill the dough for at least 30 minutes.
In a large skillet, melt the butter over medium heat. Add the kadaif and cook, stirring constantly, until golden brown and crisp, about 5 to 7 minutes. Transfer to a bowl and reserve 3 tablespoons for garnish. Let the rest cool slightly.
Melt the white chocolate in a microwave-safe bowl using short pulses, stirring between each. Stir the melted chocolate and pistachio paste into the kadaif until fully combined. Spoon the mixture into silicone molds or shape into flattened discs using a spoon. Freeze until firm.
Heat the oven to 355°F (180°C).
Scoop 2-tablespoon portions of the chilled cookie dough and flatten slightly with your hands. Place a disc of frozen filling in the center of each and fold the dough around it to enclose. Arrange the cookies on a parchment-lined baking sheet, spacing them well apart.
Bake for 8 to 10 minutes, or until the edges are set but the centers remain soft. Let the cookies cool completely.
Dip each cooled cookie partially in melted milk chocolate. Sprinkle with the reserved toasted kadaif and chopped pistachios.