In a small bowl, stir together yeast, ½ c warm water, and 1 tsp sugar. Set in a warm, draft-free place for 5-10 minutes or until very foamy.
In another small bowl, beat together flax and 3 Tbsp warm water. Set aside.
In a large bowl, sift together remaining ¼ c sugar, flour, nutmeg and salt.
Create a well in flour mixture. Add pumpkin or applesauce, margarine, flax mixture, and yeast mixture.
Stir until mixture begins to come together into a ball.
Knead dough on a lightly floured surface for 8-10 minutes, or until it's smooth and bounces back when poked gently.
Place dough in an oiled bowl and cover with plastic. Place in a warm, draft-free spot for 1 to 1.5 hours, or until doubled.
On a lightly floured surface, roll out dough until it's about ½" thick.
Using a 2.5-3" floured cookie cutter or glass, cut out circles until no dough remains.
Cover circles with a towel and let rise for another 15-20 minutes.
In the meantime, heat oil -- in a deep-fryer or deep, heavy-bottomed pot -- to 370 degrees F.*
Fry doughnuts in small batches (2-3 at a time), cooking each side for 40 seconds (or until golden).
With a slotted spoon, place cooked doughnuts on a paper towel, paper bag, or metal rack to drain any excess oil while they cool.
Repeat until all doughnuts have been fried.
Fill a pastry bag fitted with a large tip with jam.**
Set aside to cool. While still slightly warm, poke the tip of the pastry bag into the side (or bottom) of the doughnut. (If your pastry tip isn't particularly pointy, you can use a sharp knife or a toothpick to make a small opening instead.)
Squeezing pastry bag slowly but firmly, fill doughnut with desired amount of jam.
Roll filled doughnuts in powdered sugar. If you aren't patient enough to let them cool most of the way, the powdered sugar will melt and make them less picture perfect, but don't worry -- you can just roll them in sugar again once they're fully cooled.
Best eaten the day they are made.