Heat oven to 350℉/177℃.
Spray an 8 inch square pan with baking spray. Set aside.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, and nutmeg. Set aside.
With an electric mixer, beat butter and sugar until light and fluffy, about 3-4 minutes.
Add eggs, one at a time, beating well after each addition.
Stir in half of flour mixture.
Add milk.
Stir in remaining flour.
Fold in banana purée.
Spoon batter into prepared pan and bake for 31 minutes, or until cake tests done.