Heat a large (at least 5 quart) heavy-bottomed pot over medium heat with enough oil to thinly cover the bottom. Add about 3 kernels to the pot and cover with a lid. When the kernels have popped, the oil is hot enough and you can add the rest of the kernels.
Pop the popcorn until the pops are a couple of seconds apart, indicating that all or most of the kernels have popped. Remove from heat immediately and transfer the popcorn to a large pan, big enough to stir the popcorn in. Give the pan a shake to allow the un-popped kernels to fall to the bottom and remove them.
Preheat oven to 250 degrees F (95 degrees C).
Place the popped popcorn in a large oven-safe container or bowl that is large enough to allow for vigorous stirring (you can use two containers if you don't have one that is large enough. I use a large disposable metal pan (see photo above), which I wash and re-use repeatedly.
In a medium saucepan, combine butter, brown sugar, corn syrup, and salt over medium heat. Stirring, constantly, bring to a boil. Stop stirring and allow to boil for 4 minutes.
Remove from heat and quickly stir in the baking soda and vanilla. The caramel will immediately start to foam. Working quickly, pour the caramel evenly over the popcorn and stir with a rubber spatula until it is well coated and becomes harder to stir.
Bake the popcorn in the preheated oven for 1 hour, stirring every 15 minutes. With every stir you will notice that the popcorn is becoming drier. Once it becomes quite dry and no longer sticks together, remove from oven and stir again to break apart remaining clusters.