Melt butter in a skillet over medium-high heat. Stir in bananas and brown sugar. Cook for 4 minutes, stirring occasionally.
Remove skillet from heat and allow mixture to cool for 10 minutes.
Preheat oven to 350 degrees and grease a 9"x5" loaf pan.
Place the cooled banana mixture in a large mixing bowl. With an electric mixer, beat on medium speed until smooth.
In a medium bowl, whisk the flour, baking soda, and salt together. Set aside.
Combine buttermilk, vegetable oil, rum and eggs in a 2-cup measuring cup.
Add ⅓ of the flour mixture and ⅓ of the buttermilk mixture to the banana mixture and mix until combined. Continue mixing in ⅓ flour mixture and ⅓ buttermilk until everything is combined.
Place the batter in the prepared pan. Bake for 1 hour, or until a wooden pick inserted into the center comes out clean.
Cool for 10 minutes on a wire rack, then remove bread from pan. Allow bread to cool completely on a wire rack.
Making the Glaze:
In a small saucepan, cook butter over medium-high heat until it starts to brown.
Remove saucepan from heat and whisk in powdered sugar and milk until smooth.
Drizzle glaze over the bread. Allow the glaze to set up before slicing and serving.
Notes
Take care when mixing the cooled bananas. The caramelized mixture will still be warm and have a tendency to spray when beaten. I recommend wearing a bib apron during this step.
Check the ingredients in your gluten free flour mix for xanthan gum. If your flour mix doesn't contain xanthan gum, try adding ½ teaspoon to the recipe. I used Bob's Red Mill Gluten Free 1 to 1 Baking Flour which has xanthan gum in it.
Cool the banana bread completely before adding the glaze. More of the tasty brown butter glaze will adhere to the bread when it's been cooled.
Use a sheet pan under a wire rack when glazing the bread. The glaze is supposed to run down the sides of the bread and the sheet pan will keep your counter top clean.