Combine the yeast, 1.5 teaspoons sugar, and 2 tablespoons of warm water to the bowl of a stand mixer. Let it stand until the yeast foams, ~5 minutes. If it doesn't foam, start over.
Add 2 tablespoon of sugar, 2 egg yolks, coconut milk, lime juice and zest, salt, vanilla extract, and 1 cup of flour into the yeast mixture. Mix on low with the dough hook for about 2 minutes until combined.
Add the vegetable oil slowly with the mixer running. Mix well. Gradually add UP TO 1 more cup of flour. Start with ½ a cup, you might not need it all. If too sticky, a tablespoon more of flour at a time flour. You want the dough to be soft, smooth, and pull away from the sides of the bowl. Knead the dough for 3 minues to make sure it isn't sticky. If it is, add a little more flour until it isn't.
Lightly oil a medium bowl. Place the dough in the oiled bowl and cover with a kitchen towel. Let it rise in a warm place until doubled in size, about 1-1.5 hours.
In the meantime, make the curd. Whisk all the ingredients in a saucepan over medium heat. Whisk constantly until thick enough to coat a spoon, 5-8 minutes. Transfer to a bowl, cover, and chill 2 hours minimum.
Flour your surface to roll out the dough. Roll it out to ¾ inch thick. Cut into 8 squares or use a 2.5-3-inch cookie cutter*. Transfer the 8-pieces to a parchment lined baking sheet, cover with a kitchen towel, and let rise for 45-50 minutes, until ALMOST doubled in size.
Heat 3-4 inches of vegetable oil to 350F. Fry 45 seconds per side. Work in small batches and make sure your temperature is stable around 350F.
Drain on paper towels. There shouldn't be much or any excess oil if you are keeping it at 350F. Let them cool slightly before filling.
With a toothpick or skewer, make a hole in the side of each doughnut. With a piping bag** fill the doughnuts with the curd. Sprinkle with powdered sugar.
*This recipe makes 8 sufganiyot. Do not try to make larger ones or the inside will not cook completely.**If you don't have a piping bag, you can cut a ¼-inch hole in the corner of a ziploc bag.This recipe makes enough lime curd for two batches of sufganiyot. It keeps in the refrigerator in a sealed container for up to 5 days.