Place the puffed corn in an extra-large bowl and set aside. You want all 20 cups ready for the caramel.
In a large, heavy-bottomed pot set over medium-high heat, combine the butter, brown sugar, corn syrup, sweetened condensed milk, and salt. Stir gently to melt the butter and dissolve the sugar.
Bring the mixture to a boil. Continue cooking, stirring frequently, until it reaches the soft-ball stage (234-240 °F) on a candy thermometer or passes the soft-ball test when you drop a small amount of caramel into cold water.
Once the caramel is ready, remove the pot from the heat and immediately stir in the vanilla extract.
As quickly as possible, pour the hot caramel over the puffed corn. Use a large spoon or spatula to gently stir until every piece is evenly coated. Work quickly because caramel begins to set as it cools.
Let the coated puffed corn cool for about 5 minutes, just until set. The caramel will still be slightly pliable.
Once cool, break up any large clumps and transfer to an airtight container or serve immediately.