Whisk warm milk, 1 tsp of the sugar, and yeast in a bowl. Let stand 5 minutes until foamy. Add flour, remaining sugar, melted butter, egg, and salt. Mix until a shaggy dough forms, then knead on a lightly floured surface for 8-10 minutes until smooth and elastic. The dough should spring back when poked.

Place dough in a greased bowl, cover with plastic wrap, and let rise in a warm spot for 60-75 minutes until doubled in size. Press gently with two fingers; the indent should remain.

Punch down dough and roll to 1/2-inch thickness on a floured surface. Using a 3-inch round cutter and a 1-inch cutter for centers, cut out 8 rings. Re-roll scraps once. Place rings on parchment-lined baking sheets, cover loosely, and let rise 30-40 minutes until puffy.

Heat oil to 350°F in a heavy pot. Fry 2-3 donuts at a time for 90 seconds per side until deep golden brown. Adjust heat to maintain temperature. Transfer to a wire rack set over paper towels to drain and cool completely.

Process pistachios in a food processor until finely ground, almost a paste, scraping as needed. Beat mascarpone and powdered sugar until smooth. Add pistachio paste and vanilla. In a separate bowl, whip heavy cream to soft peaks, then fold into the mascarpone mixture until light and uniform. Refrigerate 20 minutes to firm up.

Stir together sugar and cinnamon in a shallow bowl. Poke a small hole in the side of each cooled donut with a skewer or chopstick. Pipe pistachio cream into both sides until the donut feels heavy and slightly firm, about 3-4 tablespoons each. Roll warm donuts in cinnamon sugar to coat completely.