Place the oil and popcorn kernels in a large 6-8 quart heavy-bottomed pot with a lid. Shake the pot to cover the kernels in oil. Place the lid on the pot and open the vent. If your lid doesn't have a vent, offset the lid a little so there's a crack in the side.
Place the pot over medium heat and wait for the kernels to start popping. Measure the sugar and salt so it's ready once the popcorn stops popping.
The moment the kernels slow and are about to stop popping, open the pot and pour in the sugar, salt, and cayenne. Turn the heat off, hold the pot by the handles, and toss the popcorn in a circular motion. Do this quickly while the popcorn and pot are still hot so the sugar melts and coats the popcorn. If your pot isn't big enough to toss the popcorn, pour it out into a large bowl to toss, but work fast. Store in an air-tight container for up to a week.