Combine brown sugar, condensed milk, corn syrup, butter, and vanilla extract in a saucepan over medium heat. Cook, stirring often, until a candy thermometer reads 234 degrees F (112 degrees C), about 5 minutes. Pour over popcorn; stir to coat evenly.
Scoop some of the popcorn mixture onto a piece of plastic wrap. Gather and twist the plastic wrap to shape into a ball. Place popcorn ball on a baking sheet lined with parchment paper; repeat with remaining mixture. Let cool until set, about 5 minutes.
