Place popped popcorn on a large baking sheet and remove any unpopped kernels or burnt popcorn. Transfer popcorn to a very large bowl and set aside.
To brown the butter, melt butter over medium heat in a large saucepan. (A light-colored pan works best for viewing the butter as it changes colors.)
Keep stirring as the butter begins to foam and brown bits form on the bottom of the pan.
Continue stirring until the butter turns from a lemony yellow to a deep golden color with a nutty aroma.
Turn the heat to low and add marshmallows. Stir until the marshmallows have completely melted.
Pour the melted marshmallow mixture over the popcorn and stir using a rubber spatula until the popcorn is evenly coated.
While the mixture is still warm, grease your hands with butter or oil and shape the popcorn mixture into balls. Press together gently but firmly enough so that they hold their shape. Do not compact them too tight, or they will be too hard to eat once set. (You can make them as big or small as you'd like. I usually do about six popcorn balls with this recipe.)
Once cooled, wrap each popcorn ball in plastic wrap and store it in an airtight container.