Warm milk in a saucepan to around 110°F (43°C). In a mixing bowl, combine the warm milk with active dry yeast and let it sit for about 5 minutes until it becomes foamy. Whisk in the egg and softened butter, then gradually add flour, sugar, and salt, mixing until a shaggy dough forms. Knead the dough on a floured surface for 5–7 minutes until smooth and elastic.

Place the kneaded dough into a lightly oiled bowl and cover with a damp cloth. Set it in a warm area to rise until doubled in size, about 1 hour.

In a medium bowl, mix the pistachio butter, brown sugar, and ground cinnamon until smooth. Set aside.

Once the dough has risen, gently punch it down and roll it out into a rectangle, about ¼ inch thick. Spread the pistachio filling evenly over the dough, leaving a small border. Roll the dough tightly from one end and slice into 4–6 pieces.

Arrange the sliced rolls in a greased baking dish and cover with a kitchen towel. Let them rise again until puffy, about 30–45 minutes.

Preheat the oven to 350°F (175°C). Once heated, bake the rolls for 23–30 minutes until golden brown.

In a bowl, whisk together powdered sugar, vanilla extract, heavy cream, and pistachio butter until smooth. Adjust consistency as needed.

Allow the rolls to cool slightly before icing. Spread the creamy pistachio icing on top and serve warm.