Activate the yeast: In a small bowl, combine the warm milk, sugar, and warm water. Sprinkle the yeast over the top and let it sit for 5 minutes until frothy.
Prepare the wet ingredients: In a large mixing bowl, combine the mashed bananas, softened butter, egg, and vanilla extract.
Combine dry ingredients: In another bowl, sift the flour and salt together.
Mix the dough: Gradually add the dry ingredients into the wet ingredients, stirring until the dough begins to form.
Knead the dough: Turn the dough out onto a floured surface and knead for about 8-10 minutes until smooth and elastic.
Let the dough rise: Place the dough in a greased bowl, cover with a clean towel, and let it rise for 1-2 hours, or until it doubles in size.
Shape and proof the dough: Once risen, punch the dough down gently to remove air bubbles, then shape it into a loaf.
Bake the bread: Preheat the oven to 350°F (175°C). Place the dough in a greased loaf pan and bake for 30-40 minutes, or until golden brown and a toothpick inserted into the center comes out clean.
Cool and serve: Let the banana bread cool in the pan for 10 minutes before transferring to a wire rack to cool completely.