Heat up the water (I just microwave it) to slightly over body temperature. When you stick your finger in, it should feel warm, but not hot or scalding.
Mix the sugar and yeast into the water to dissolve. Let the yeast proof for about 5 minutes.
In a separate bowl, mix the sugar, oil, and mashed banana very well. Do not blend the banana because it will lose its stickiness. The banana is acting like a binding agent (kind of what eggs do).
Using your electric mixer with the dough hook, or a wooden spoon, combine the yeast mixture, banana mixture, and the flour 1 cup at a time. Knead for about 5 minutes. If you do not have a electric mixer, knead by hand for about 10 minutes. The dough should be tacky and smooth. Start with 3 cups of flour then add more as needed.
Once the dough is smooth and when you poke it, it doesn't stick to your finger, brush the dough with a bit of oil and cover the bowl with a damp cloth.
Let the dough rise in a warm place until double in size. It will take about 40 minutes to an hour.
When the dough is done rising, form your brioche. I have tried a braid and just a normal blooming loaf, but do it any way you want.
Let the brioche rise for another 10-20 minutes. I like letting my breads rise in my oven that is off with the light on. The oven prevents any dust or dog fur, in my case, from getting on the dough.
Bake at 350 degrees Fahrenheit for 30 minutes. The bread should be a golden brown and when you tap the top, it should sound hollow. Let cool slightly and then enjoy!