Have a hot pad or kitchen towel and a paper grocery bag or large baking sheet handy. Heat the vegetable oil in a large heavy-bottomed Dutch oven or pan over medium heat. Place a couple of kernels of popcorn in the pan. When the kernels pop, proceed to the next step.
Once the oil is hot enough, add the butter and stir quickly into the oil to prevent burning. Immediately add the popcorn and stir (the butter may not be completely melted). Add the sugar right away and stir again. Cover the pan, and start shaking it gently back and forth on the burner, using a hot pad or kitchen towel to protect your hands. Keep shaking as the popcorn pops.
Once you start hearing 1 or 2 seconds of quiet between the pops, turn the heat off immediately. This means the popcorn is mostly popped. (Be careful not to overcook, as the popcorn can easily burn.) Pour the popcorn into the paper bag or onto the baking sheet and sprinkle with salt, cinnamon, and cayenne. Allow to cool a few minutes before placing in a bowl. (Careful, the hot sugar makes the popcorn extremely hot!)