Preheat oven to 350 degrees. Grease muffin tins.
In a saute pan, melt butter over low-medium heat. Add mashed bananas and sugar. Stir and bring to a boil. Cook for 3-5 minutes or until begins to caramelize.
Remove from heat. Pour into a heat safe bowl and allow to cool slightly.
Whisk in egg and vanilla.
Add flour, baking soda, and salt. Stir just until combined.
Pour into prepared muffin tins - 9 for big, 12 for small muffins.
Top with coarsely ground sugar, if you want. I highly recommend - if you don't have coarse ground sugar, a little brown sugar will do too!
Bake 20 minutes for 9 muffins, less for 12 muffins. Or until golden brown.