Preheat the oven to 350 degrees F and move a rack to the middle of the oven. Line a baking sheet with parchment paper.
Heat the oil in large heavy-bottomed pot with a lid over medium heat. Add the popcorn kernels and place the lid slightly ajar on top of pot. Cook until most of the kernels pop, 7 to 10 minutes (the popping sounds slow down when it's almost done). This will make 10 to 12 cups of popcorn.
While the popcorn is popping, melt the butter, honey, vanilla and salt in a small saucepan over low heat until the mixture is uniform. Transfer the popcorn onto the prepared baking sheet. Discard any unpopped kernels that settle at the bottom of the pot. Pour the butter mixture evenly over the popcorn. Toss the popcorn with two spatulas until coated. Bake, stirring every 5 minutes, until the popcorn is a deep caramel color, 20 minutes.
Remove the baking sheet from the oven and cool for 5 minutes. Wearing clean plastic gloves, use your hands to press the popcorn into 2-inch balls. Towards the end you may need 1/2 to 1 teaspoon of extra honey to act as the "glue" for the balls.
You can substitute the popcorn kernels with one 3.5-ounce bag microwave popcorn prepared according to the package instructions.
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