Heat the oven to 350ºF. Coat an 8-by-4-inch loaf pan with cooking spray.
Whisk together the flour, salt, baking soda, baking powder, cinnamon, and nutmeg.
Beat the peanut butter, butter, and sugar with a mixer on medium-high speed until light and fluffy, 2 to 3 minutes. Beat in the eggs one at a time on medium speed, and then beat in the vanilla extract. Beat in the flour mixture on low speed. Fold in the mashed bananas.
Spoon the batter into the prepared pan and smooth the top. Bake until a toothpick inserted in the center of the bread comes out clean, 50 to 60 minutes. Cool the bread in the pan for 10 minutes; then remove it to a rack to cool completely.
Tip: To make mini-loaves, pour the batter into three 5-by-3-inch loaf pans and reduce the cooking time by 15 to 20 minutes.