Preheat the oven to 350 degrees F and place a rack in the center of the oven.
In a large Dutch oven or heavy pot with a lid, warm the oil over medium heat, then stir in the popcorn kernels. Cover the pot and shake periodically until the kernels start to pop, then shake more until the popping slows. This should take a total of 8 to 10 minutes. Remove the lid, then spread the popcorn evenly on a baking sheet and sprinkle with the peanuts if using, and set aside.
In a small saucepan set over medium heat, melt the butter, then stir in the sugar and cook until the mixture starts to bubble and the sugar dissolves, 4 to 5 minutes. Drizzle over the popcorn and peanuts. Bake the popcorn until the sugar crisps and the popcorn has a light golden color, 10 to 15 minutes.