Heat oven to 250° F (121ºF) with rack set in middle position. Line a baking sheet with parchment paper.
Heat 1 ½ tablespoons oil in a large pot over medium-high. Add 5 kernels of corn and allow to heat until a kernel or two pops. Add ¼ cup corn kernels, cover pot with lid and give it a shake to coat kernels in oil. Cook, shaking pot occasionally, until kernels start to pop. Continue shaking pot frequently as corn pops until popping slows to about 3 seconds between pops. Remove pot from heat and uncover; cool while making caramel.
In a medium saucepan set over medium-high heat, combine ½ cup packed brown sugar, 5 tablespoons butter, 2 tablespoons corn syrup and 2 heaping teaspoons gochujang. As mixture heats, whisk to combine. Bring to a boil and cook, without stirring, until mixture turns deep golden brown and registers 250–255ºF on a thermomter, about 3 minutes.
Remove pan from heat and whisk in 1 teaspoon vanilla, ½ teaspoon salt and ¼ teaspoon baking soda.
Drizzle caramel over popcorn and, using a rubber spatula, stir quickly to evenly coat.
Transfer popcorn to prepared baking sheet and spread into an even layer. Bake, stirring every 20 minutes, until caramel has slightly darkened and popcorn is nearly dry to touch, about 1 hour. Let caramel corn cool completely on sheet pan, at room temperature, before breaking apart (as needed) and serving.
Store in an airtight container or zipper-lock bag for up to 2 weeks.