Place the yeast, warm water and 1 tsp granulated sugar into the bowl of an electric mixer fitted with the dough hook attachment. Allow this to sit for 10 minutes, until the yeast begins to develop.
Add the flour, ½ cup sugar, eggs, butter and salt. Set the mixer to medium speed and knead for 8-9 minutes, until the dough is smooth, soft and bounces back when poked with your finger.
Place the dough in an oiled bowl and cover with plastic wrap. Place in a warm spot until doubled in size, about 1 hour.
Roll the dough out to ¼ inch-thick and cut into 4 inch-wide circles with a cookie cutter. You should be able to get 10 doughnuts. Transfer the doughnuts to a lightly floured baking sheet and cover with a sheet of plastic wrap. Place in a warm spot and let rise for 20 minutes.
Pour about 5 inches of canola oil into a pot and set to medium heat. Attach a deep-fry thermometer and heat the oil to 370F. Fry 1 or 2 doughnuts at a time, cooking for 1 minute on each side or until golden.
Remove the doughnuts from the oil with a slotted spoon and place on a baking sheet lined with paper towel. Cool completely.
Poke a hole into the sides of the doughnuts with a butter knife. Place the pineapple jam in a piping bag and squeeze into the doughnuts.