Preheat the oven to 375°F (190°C) and line two baking sheets with parchment paper or silicone baking mats.
In a large bowl, beat 0.5 cup unsalted butter, 0.5 cup granulated sugar, and 0.25 cup packed light brown sugar until light and fluffy, about 2–3 minutes.
Add 1 large egg and 1 teaspoon vanilla extract to the butter–sugar mixture and beat until combined.
Stir in 1 cup mashed ripe banana until just incorporated and slightly lumpy.
In a separate bowl, whisk together 1.5 cups all-purpose flour, 1 teaspoon baking soda, and 0.5 teaspoon salt.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed or by hand until just combined.
Drop rounded tablespoons of dough about 2 inches apart on the prepared baking sheets.
Bake for 10–12 minutes, until the edges are lightly golden and centers spring back when pressed.
Let cookies cool on the baking sheets for 3–5 minutes before transferring them to a wire rack to cool completely.