Using an electric mixer with a bread hook, place the warm water (just barely over luke-warm, too hot will kill the yeast) and sugar in the mixing bowl. Sprinkle the yeast over the water and let it foam for 10 minutes.
Then turn on the mixer and add the apple butter, egg and salt. Slowly add the flour. Once all the flour is in the bowl, add the melted butter. "Knead" the dough with the bread hook for 5-8 minutes.
The dough will be smooth, but very tacky. Dump it out onto a floured work surface. Using a floured knife, cut the dough into 4 quarters. Then cut each quarter into 6 equal pieces, to make 24 small dough segments.
Turn the ends of each dough segment under to create little balls. Then place a large piece of parchment paper in a 6-quart slow cooker, and press down. Arrange 12 dough balls in the slow cooker. Then carefully lift the parchment paper out, and repeat with the remaining dough balls and another piece of parchment, creating two separate batches. Place the lid on the crockpot and cover the additional dough balls on the counter, with plastic wrap. Allow them to rise for 1 hour at room temperature. *At this point the dough balls could be put in the fridge for up to a week, until ready to bake.
Sprinkle the rolls with a little brown sugar. Turn the slow cooker on high and "bake" for 60-90 minutes, depending on your crockpot. To test, touch the top of the rolls. If they feel slightly firm and are no longer tacky, the rolls are ready. Repeat with the second batch of dough balls. Each batch will cook for 60-90 minutes, 3 hrs total.