Pour the infused lilac sugar into a wide, shallow medium bowl. Set aside.
When you're ready to fry the donuts, dust the surface of the refrigerated dough with flour and use kitchen scissors to cut off a 1-pound portion. The portion should be around the size of a grapefruit. Dust the portion with more flour and quickly shaping it into a ball by stretching the surface of the dough around to the bottom on all four sides, rotating the ball a little bit as you go.
Roll the dough into a 1/2-inch thick rectangle on a lightly floured surface. Using a 3-inch donut cutter, stamp out 3-inch diameter rounds with 1-inch diameter holes. Allow the dough to rest on a lightly floured surface for 15 minutes.
Meanwhile, fill your deep-fryer or cast iron skillet with at least 3 inches of oil. Bring the oil to 360 (F) to 370 (F), as determined by the candy thermometer.
Carefully drop your stamped donuts into the hot oil, two or three at a time depending on the size of your cast iron skillet. Be sure to leave plenty of room for the donuts to float to the surface. Do not overcrowd, or they will not rise nicely. Fry for 1 to 2 minutes per side, or until light golden brown. Remove with a slotted spoon, draining the donuts of grease on a wire rack lined with a paper towel.
Allow each donut to cool for a minute, then, while each donut is still hot, transfer them one by one into the bowl of lilac sugar and toss until completely coated. Allow the donuts to rest for a few more minutes before serving/eating. Be sure to eat while warm since they are best on the day they are made!