Add sugar and yeast* to the lukewarm water in a large bowl. Allow it to sit for about 10 minutes or until the yeast starts to bubble. Pro tip: Water temperature should not be above 110°F, or it will kill the yeast. Aim for 95º-105ºF, measuring with a thermometer for accuracy

Meanwhile, in a small bowl, beat the eggs. Cut the butter into small cubes and set aside.

After 10 minutes, add flour, eggs, and vanilla to the yeast mixture in the bowl of a stand mixer. Using a stand mixer with the dough hook attachment, knead well for about 5 minutes. Alternatively, knead by hand on a clean counter.

After about 5 minutes, add the salt and knead until incorporated. Slowly incorporate the butter cubes, one cube at a time, and continue kneading. This step takes a little while. It is tempting to add the butter faster or add it all at once. Resist this urge and continue adding the cubes one at a time. If the butter seems like it's not incorporating well with the stand mixer and dough hook, remove the dough and place it on a clean surface. Knead it by hand, using a dough scraper to pick up and add in any pieces of dough that stick to the counter. This may seem wet and sticky and messy for some time, but be patient. It all comes together.

After all the butter has been incorporated, the dough will be glossy and smooth. Place the dough in a clean oiled large bowl and cover the bowl with plastic wrap. Allow the dough to proof at room temperature for about 2 hours, or until doubled in size.

Punch down the dough.*If you are making the dough ahead of time and plan to fry the donuts the next day, re-cover the bowl after punching it down and place in the fridge to chill for 12 - 16 hours or overnight

Turn the dough out onto a clean counter and cut it into 70g pieces - you should get 14 pieces. If you do not have a kitchen scale, you can eyeball it. Shape each piece into a smooth, tight ball and place on a baking tray lined with parchment paper and dusted lightly with flour. Leave plenty of room between the donuts to allow room for them to rise. Cover the donuts lightly with plastic wrap and leave to rise for about 2 hours, or until about doubled in size

In a large pot (dutch oven or cast iron) filled halfway with vegetable oil (this should be about 3-4 inches of oil, enough to allow the donuts to float and not sit on the bottom of the pot), clip a candy thermometer or deep-fry thermometer to the side and heat over low until the oil reaches 350°F. Carefully remove the donuts from the baking tray, taking care not to deflate them, and place them gently into the oil using a long-handled spider tool or slotted spoon. Fry for three minutes on one side then carefully flip with the slotted spoon and fry for 2-3 minutes more on the second side until golden brown. Keep an eye on the temperature of the oil to ensure it doesn't get too hot or too cool - adjust heat as necessary. Remove donuts from the oil with the spider tool or slotted spoon and place on a paper towel-lined baking sheet to absorb the excess oil; I typically line the baking sheet with at least two layers of paper towels.

Once the donuts have cooled slightly (after about 1-2 minutes), place ½ cup granulated sugar in a dish and roll each donut in the sugar. Set aside to cool completely before filling

If you have a plastic condiment bottle or squeeze bottle, fill it with jam. You may have to cut the tip a bit to allow the thick jam to pass through, especially if it has small pieces of fruit in it. Otherwise, fill a piping bag affixed with a large round piping tip with the jam/jelly. Using a butter knife or long skewer, cut a small hole in the side of each donut about ¾ of the way into the donut and pipe a generous amount of jelly into the center of the donut. Eat the donuts right away or store in an airtight container for up to three days