In a medium bowl, use your hands to toss together 1 cup granulated sugar and fresh lilac flower petals. Use your fingers to gently rub the petals and sugar together to help release oils from the flowers. Transfer to a mason or glass jar, seal, and shake vigorously. Store in a cool, dark place for at least one night (but preferably 3 nights total), giving the jar a shake every now and then.

Sprinkle 3 3/4 cups all-purpose flour over the wet ingredients, before turning the mixer on to its lowest speed and mixing in the flour until just combined. You may notice lumps in the dough, but don't worry — these will disappear in the finished product.

Cover the bowl loosely with a flour cloth or a plastic wrap (it's important to let this dough breathe a little, so your cover doesn't have to be airtight) and allow the dough to rest at room temperature for 2 hours, before covering tightly with plastic wrap and refrigerating.

The dough can be used as soon as it's thoroughly chilled, at least 3 hours — it's a little hard to work with when it hasn't chilled properly, so be sure to chill it for as long as it needs to get it to firm up. You can refrigerate the dough for 5 days; after 5 days, divide the dough into 1-pound portions in an airtight container and transfer it to the freezer. The dough will keep for up to 2 weeks in the freezer. When using frozen dough, thaw in the refrigerator for 24 hours before using, before allowing the usual rest and rise times.

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.

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!