Preheat the oven to 350 degrees F (175 degrees C). Line a 12-cup muffin tin with paper liners, or grease cups. Whisk together flour, salt, baking powder, and baking soda; set aside.

Beat butter, peanut butter, and sugar together in a bowl with an electric mixer or whisk until light and fluffy. Add bananas and beat again until light and fluffy. Mix in eggs one at a time; stir in milk and vanilla.

Add flour mixture; mix on low speed until about 75% of flour is incorporated. Use a spatula to stir in chocolate chips, and to fully incorporate flour. Divide batter evenly between prepared muffin cups.

Bake in the preheated oven until a toothpick inserted near the center comes out clean, 23 to 25 minutes.

Cool muffins in the tin for about 15 minutes; transfer to a wire rack to cool completely, about 15 minutes more.

For icing, combine peanut butter, milk, and powdered sugar in a bowl and stir until a thick icing is formed. Add more powdered sugar as needed for an icing thick enough to hold its shape when piped. Add icing to a piping bag and pipe in a zigzag pattern over cooled muffins.