Make the dough: In a medium bowl, combine the yeast, milk, water and 1 teaspoon granulated sugar and give it a little stir with a wooden spoon. Let it sit about 5 minutes, until it becomes foamy on top.
In a stand mixer fitted with a dough hook, mix the flour, salt, cardamom, cinnamon, orange zest and remaining 1/4 cup granulated sugar. Add the beaten egg and yolks, vanilla and yeast mixture and mix to form a very stiff dough. (It may be hard to incorporate at first, but keep mixing.)
With the mixer on, gradually add the butter, piece by piece. This, too, will seem like it isn’t going to incorporate, but keep on mixing for about 8 to 10 more minutes, scraping down the dough hook occasionally, until your dough is smooth and slightly sticky.
Transfer the dough to an oiled bowl, cover it with plastic wrap and let it sit at room temperature until it has doubled in size, about 2 hours. (Or refrigerate the dough for up to 2 days; bring to room temperature before rolling out.)
Cut out the doughnuts: Turn out the dough onto a lightly floured work surface. Using a rolling pin, roll it out until about 1/2 inch thick.
Cut out rounds of dough using a 2 1/2-inch round cookie cutter. Press the scraps of dough together and let sit 10 minutes, then reroll the scraps and cut out more rounds. (At this point, you can refrigerate the rounds up to 3 hours before frying.)
Transfer the rounds to a baking sheet lined with parchment paper. Cover with plastic wrap or a dish towel and let rise for another hour, or until doubled in size.
Fry the doughnuts: In a large cast-iron pot fitted with a deep-fry thermometer, heat 3 to 4 inches of vegetable oil to 350˚. Fry the doughnuts in batches of 3 or 4 for 1 1/2 minutes on each side until deep golden brown.
Using tongs, remove the doughnuts from the hot oil and transfer to a rack set on a baking sheet to drain and cool.
Make the filling: In a small bowl, combine the jam and rose water. Taste and adjust the proportions as desired. Spoon the jam into a squeeze bottle or piping bag fitted with a 1/4-inch tip.
Make the glaze: In a medium bowl, whisk the confectioners' sugar, milk, vanilla and salt until smooth. It should be the consistency of a slightly thick glue — if it’s too thick, add a little more milk; if it’s too thin, add a little more confectioners’ sugar.
Dip the cooled doughnuts in the glaze and let any excess drip off. If the glaze has a hard time sticking to the doughnuts, that means it’s too thick and that you should add more milk.
Fill the doughnuts: Stick the squeeze bottle or pastry bag into the center of a doughnut, going as far down as you can without puncturing the other side. Wiggle it around, then fill up the hole with jam, leaving a circle of jam on top of the glaze. Sprinkle with pistachios and rose petals. Serve any remaining jam on the side for dipping.