For the Pistachio Cream Filling:

• 3 tablespoons unsalted butter or coconut oil, divided
• 5 ounces coarsely chopped kataifi pastry dough (about 2 packed cups), thawed if frozen
• 1 (7- to 8-ounce) jar creamy pistachio butter or pistachio cream
• 2 tablespoons well-stirred tahini
• 1/4 teaspoon kosher salt

For the Milk Chocolate Coating:

• 3 tablespoons white chocolate chips that contain cocoa butter, such as Guittard
• 1 to 4 drops liquid or gel green food coloring (optional)
• 12 ounces good-quality milk chocolate chips (about 2 cups), divided