Preheat the oven to 350 degrees. Prepare your loaf by spraying with pan release or buttering the pan with softened or melted butter.
Mix the flour, baking soda, and salt in a small bowl and set aside.
In another medium-sized bowl, add in the bananas. Mash them with a fork. Next, add in the brown sugar, butter, and oil. Mix well. To this add the egg, vanilla, and Greek yogurt.
Now add the grated carrots and zucchini. Make sure you squeeze out the zucchini to get out excess moisture. Mix and add the dry ingredients.
Mix the wet and dry together, be sure not to overmix. You only want to mix this until they come together.
Pour the batter into the prepared pan and move to the preheated oven. Bake for 50-55 minutes.
Remove the pan from the oven. Let it sit in the pan for 5-7 minutes. Turn it out onto a cooling rack.
That is it. The loaf is done. You can enjoy this bread warm, cool, or even cold! This bread is amazing any way you choose to eat it!