Add the oil and one kernel to a 6-quart or larger pot. Place the lid on the pot and turn the heat onto medium-high. Wait for the test kernel to pop. Once the kernel pops, add the rest of the kernels, replace the lid, and give the pot a swirl to coat the kernels in the hot oil. Wait for the kernels to begin to pop.
Once the kernels begin to pop rapidly, crack the lid a little bit to allow excess steam to escape. Pop the rest of the kernels. Continue popping until they slow to about one pop every other second. Once popped, transfer the popcorn to a large bowl and set aside.
Add the sugar, corn syrup, and salt to a separate medium saucepan and heat over medium-high. Bring the mixture to a boil and stir to dissolve the sugar. Once the pot comes to a boil, boil the sugar mixture for 2 1/2 minutes.
Next remove the saucepan from the heat and add the butter, vanilla extract, and marshmallows. Stir quickly to melt the ingredients into the sugar mixture.
Once everything has melted, pour the sugar mixture over the cooked popcorn. Mix together quickly with a rubber spatula to coat the popcorn.
The popcorn will be slightly hot from the hot sugar mixture, so allow it to cool slightly. Once it's slightly cool enough to touch, rub a dab (about 1/4 tsp) of cooking oil into your hands, grab a small handful of popcorn, and start to form 3-inch round popcorn balls. Place each popcorn ball on a parchment lined baking sheet to finish cooling and to set.
Be sure to work quickly because if the sugar mixture cools too much, it will be hard to form the popcorn balls. You should be able to make 12 3-inch popcorn balls. Serve at room temperature and enjoy!