BREAD
Preheat oven to 350°. Place a rack in the center of the oven. Spray a 9 x 5 loaf pan with baking spray (or butter and flour) and set aside.
Melt 3 tablespoons butter in a large skillet over medium-high heat. Add brown sugar and bananas; sauté 4 minutes, stirring occasionally. Remove from heat; cool 10 minutes. Place banana mixture in a large bowl.
Beat with a mixer at medium speed until smooth.
Combine buttermilk, canola oil, rum and eggs (or flax eggs). Weigh or lightly spoon flour into dry measuring cups; level with a knife. In a large bowl, whisk together flour, baking soda, and salt. Add flour mixture and buttermilk mixture alternately to banana mixture, beginning and ending with flour mixture; beat at low speed just until combined. Scrape batter into a 9 x 5-inch metal loaf pan coated with baking spray.
Bake for 1/2 on hour at 350 then lower the heat to 325 and bake for about another 20-30 minutes, or until a skewer inserted in the center comes out clean. Every oven is different. I like to lower heat halfway through baking time so that it doesn't bake up too quickly and brown too fast.
GLAZE
Melt remaining 1 tablespoon butter in a small, heavy saucepan over medium-high heat. Cook 3 minutes or until butter begins to brown; remove from heat. Add powdered sugar and milk, stirring with a whisk until smooth. Drizzle glaze over bread. Let stand until glaze sets.