Preheat the oven to 250 degrees F.
First make the popcorn: Put a large pot over medium heat, and add the oil and popcorn. Place a lid on the pot, slightly askew, so that steam can escape. Pop the corn until there are a few seconds between pops, and it sounds like they are mostly cooked.
Pour the popped corn into a large roasting pan (like one you would use to roast a turkey) and sprinkle with kosher salt.
In the same pot that you made the popcorn, melt the butter, brown sugar, corn syrup, peanut butter and salt. Bring to a boil, stirring, for 7 minutes.
In a tiny bowl, mix the baking soda and vanilla together until the baking soda is mostly dissolved.
Remove the pot from the heat, and stir in the vanilla and baking soda mixture. The caramel should be very light and frothy. Immediately pour the caramel over the popcorn, and gently fold the caramel into the popcorn to coat it. (At this point, you might think there is not enough caramel for the corn, as there will be globs of caramel, and a lot of the popcorn will not seem well-coated. But don't worry! There will be plenty.).
Bake the caramel corn for 15 minutes. Remove the roasting pan from the oven and gently stir the popcorn into the melting caramel until the popcorn is better coated. Return to the oven for another 15 minutes, after which the popcorn should be evenly coated with the caramel.
Return the caramel corn to the oven and bake for a final 15 minutes. Remove from the oven and give the corn one final stir to coat thoroughly. Let cool completely, about 30 minutes.
When the popcorn is totally cool, it will have a nice, thin, crispy coating. Serve immediately or store in airtight containers for up to 10 days.