Preheat oven to 350°F (175°C). Line a baking sheet with parchment.
Whisk flour, baking powder, and salt in a bowl.
Cream butter and brown sugar until smooth, then beat in mashed banana, egg, and vanilla.
Mix in dry ingredients until just combined, fold in walnuts (and chips if using).
Shape dough into a log about 12 x 3 inches on the sheet. Flatten slightly.
Bake 25–30 minutes until golden. Let cool 10 minutes, then slice 3/4-inch thick.
Arrange slices cut-side down and bake 10–12 more minutes per side until crisp. Cool completely.