Combine the yeast and the warm water in the bowl of a stand mixer fitted with a paddle attachment and allow to sit for 6 to 7 minutes, until foamy.
Sift together the flour, sugar, and salt into a very large mixing bowl and set aside.
Add the milk, melted butter, vanilla bean paste and orange zest to the mixing bowl and mix at medium-low speed until fully combined.
Add the eggs to the mixture, all at once, and mix at low speed until fully combined, about 2 minutes.
Add 2 to 2½ cups of the flour mixture and mix at low speed until fully combined. Stop the mixer and scrape the bottom and side of the mixer bowl as well as the paddle. Switch to the dough hook. Add the remaining flour mixture and mix at low speed to combine fully. Scrape the bottom and side of the bowl again. Continue to mix at low speed, kneading the dough for 6 to 7 minutes, until it is shiny, smooth, and tacky.
Oil a bowl and scoop the dough into it an oiled bowl, cover lightly with plastic wrap and a kitchen towel, and refrigerate for 2 hours or until it increases in volume by about 25 percent.
While the dough is resting and gently rising, make the fig jam: In a medium saucepan, combine the Orchard Choice or Sun-Maid California Figs, orange juice, pomegranate molasses, brown sugar, cinnamon stick, salt, and nutmeg and set over medium-low heat, and cook, stirring, until the mixture is fully combined about 2 to 3 minutes. Lower the heat to a simmer and cook, uncovered, stirring occasionally, for about 45 minutes while you continue making the doughnuts. When the jam is done, remove the cinnamon stick. Remove the pot from heat and allow to cool. It will thicken as it cools. You will have about ¾ to 1 cup of jam.
Line 2 rimmed baking sheets with parchment paper. Dust a work surface and rolling pin lightly with flour.
When the dough has risen, scoop it onto the work surface. (After chilling, it will be easy to handle.) Cut the dough in half. Wrap one portion in plastic wrap and refrigerate. Roll out the other into a 13- by 6½-inch rectangle that is about ¾ inch thick. Dip a 3- to 3¼-inch round cookie cutter into some of the work-surface flour, and cut out 8 circles. Place any scraps of dough into a plastic bag into a plastic bag, seal, and refrigerate.
Dust any flour off the doughnuts. (Excess flour will burn quickly when frying.) Transfer them to the prepared baking sheets. Cover with kitchen towels and allow to rise for 15 minutes, or until they increase in volume by about 25 percent.
Pour 8 to 9 cups of oil into a deep, wide 6-quart Dutch oven. The oil should be at least 2½ inches deep and no more than half way up the side. Set a frying thermometer into the oil, making sure it does not touch the bottom of the pot. (If you are using an instant-read thermometer, position it near the stovetop.) Line another baking sheet with paper towels and set it nearby. Set a cooling rack over more paper towels in the work area.
Heat the oil over medium heat until it reaches 355°F to 360°F. If it is too hot, lower the heat to a simmer; if it is not quite hot enough, raise the heat. Maintaining the temperature within a few degrees up or down is extremely important; the doughnuts should cook through, but not absorb too much oil and get greasy.
Using a slotted spoon or kitchen spider, gently place 4 doughnuts, one at a time, into the hot oil. The doughnuts will begin to expand as soon as they touch the oil. After about 1½ to 2 minutes they will be golden brown on the bottom. Turn them with the slotted spoon or spider and fry the second side, about 1½ to 2 minutes. If you have an instant-read thermometer, the internal temperature should be 195°F to 200°F. Carefully transfer the doughnuts to the towel-lined platter. Gently place the remaining 4 doughnuts into the oil and repeat the frying process. When the first 4 doughnuts are cool enough to touch, transfer them to the cooling rack. When the second batch is fried, transfer to the towel-lined platter to cool. Turn the heat off from under the oil. Let the first batch of doughnuts cool completely—at least 30 minutes—while you make the second batch.
Remove the dough from the refrigerator. Roll and cut it and allow to rise as before, adding any scraps to the plastic bag in the refrigerator. While the doughnuts are rising, reheat the oil to 355°F to 360°F. It will reheat quickly. Fry, drain, and cool the second portion of doughnuts, as before. Turn the heat off after frying the last doughnut.
Place the refrigerated scraps from the first two batches on the work surface and knead together. Roll the dough into a 6½-inch square, and cut 4 more doughnuts. Cover with a kitchen towel and allow to rise for 15 minutes. Reheat the oil to 355° to 360°F. Fry, drain, and cool, as before.
Scoop the jam into a pastry bag fitted with a small, sharp tip. Insert the tip into the side of a cooled doughnut and squeeze for 2 to 3 seconds, or until you see a small dab of jam at the point of entry. Place the filled doughnut back on the cooling rack or on a clean plate and repeat with all of the doughnuts, working with the second and third batches after they have cooled.
If you wish, you can liberally dust the filled doughnuts with confectioners’ sugar or you can make the sour cream glaze.
Combine the confectioners’ sugar and sour cream in a mixing bowl and whisk until completely smooth. Dip each doughnut into the glaze and place on the rack to let the glaze harden, about 30 minutes.