Generously grease a 9x13" baking pan with butter or a neutral-flavored cooking oil. Set aside.
Using an air popper or other popcorn popping method of your choice, pop approximately 1/2 cup of plain popcorn kernels. Measure out 16 cups of popcorn and set it aside (if you're short of 16 cups after 1/2 cup, pop a little more - many popcorn poppers can only accommodate 1/2 cup per batch). Pick out any unpopped kernels of popcorn that you see.
Melt 1/2 cup butter and 1/2 teaspoon kosher salt together in a large bowl in the microwave on high temperature in 30 second intervals (alternatively, you can do this in a large saucepan or Dutch oven on the stovetop).
Once the butter is mostly melted, stir in 400 grams marshmallows. Melt on high speed in 30 second intervals again until the marshmallows are melted. At this point, the butter and marshmallow will be separated, but stir it vigorously for at least 30-60 seconds, and the mixture will come together and emulsify. (Note that if the mixture still looks greasy with visible butter, it's not done - keep stirring). Stir in 1 teaspoon vanilla extract.
Add the popcorn to the bowl, and stir carefully to avoid crushing the popcorn until the popcorn is evenly coated with the marshmallow mixture. Pour the popcorn mixture into the prepared baking pan and gently press it into the pan. Let the pan set at room temperature for 30-60 minutes.
Cut into squares of your desired size (remove the whole slab from the pan before cutting to avoid damaging your pan). Store the squares in an airtight container or zipper bag at room temperature for up to 2 days.