How to Make It
In a small microwavable bowl, warm the milk at high power to 110°, 1 minute. Sprinkle the yeast over the milk and let stand until foamy, about 10 minutes.
In a stand mixer fitted with the dough hook, combine the flour with the superfine sugar. Add the milk mixture, the butter, egg, Armagnac, vanilla and salt. Knead at low speed, scraping down the side of the bowl, until the dough starts to come together, about 4 minutes. Increase the speed to medium and knead until the dough is soft and pulls away from the bowl, about 10 minutes.
Lightly brush a large bowl with oil. Place the dough in the bowl, cover with plastic wrap and let stand in a warm spot until doubled in size, about 2 hours. Lightly brush 2 rimmed baking sheets with oil. Punch down the dough and cut in half. Roll each half into a 12-inch log. Cut each log into 12 equal pieces, roll into balls and arrange on the prepared baking sheets, about 3 inches apart. Cover with plastic wrap and let stand until doubled in size, about 1 1/2 hours.
In a medium bowl, combine the granulated sugar with the cinnamon.
In a large saucepan, heat the 5 cups of oil over moderate heat to 325°. Working in batches, fry the doughnuts, turning once, until golden, about 2 minutes per side. Using a slotted spoon, transfer the doughnuts as they’re done to the cinnamon sugar and toss to coat. Let cool on a rack, about 15 minutes.
Spoon the jam into a pastry bag or small resealable plastic bag with 1 corner snipped. Pipe about 2 tablespoons of jam into 2 opposite ends of each doughnut and serve as soon as possible.