In a small bowl, whisk together the warm milk, granulated sugar, and yeast. Let stand for 5 to 10 minutes until foamy.

In a large bowl, combine the flour and salt. Add the yeast mixture, softened butter, egg, and vanilla extract. Stir until a soft dough forms, then turn out onto a lightly floured surface and knead for 8 to 10 minutes until smooth and elastic.

Place the dough in a lightly greased bowl, cover with plastic wrap or a damp towel, and let rise in a warm spot for about 1 hour, or until doubled in size.

While the dough rises, make the pistachio cream. In a food processor, grind the pistachios until finely ground. Add the heavy cream, powdered sugar, and vanilla extract, and process until smooth and creamy. Transfer to a bowl, cover, and refrigerate until firm, about 20 minutes.

Once the dough has risen, roll it out on a floured surface to about 1/2-inch thickness. Use a round cookie cutter (about 3 inches) to cut out circles, then use a smaller cutter (about 1 inch) to cut out the centers, forming rings. Place the rings on a parchment-lined baking sheet, cover, and let rise for 30 minutes.

Heat 2 to 3 inches of oil in a deep pot or Dutch oven to 350°F. Fry the donut rings in batches, cooking each side for 1 to 2 minutes until golden brown. Transfer to a paper towel-lined plate to drain excess oil.

Let the donuts cool slightly, then use a piping bag fitted with a small tip to fill each donut with the chilled pistachio cream. Dust with powdered sugar and sprinkle crushed pistachios on top. Serve immediately.