Preheat oven to 350°F and line a baking sheet with parchment paper.
In a small bowl, make the streusel by mixing flour, brown sugar, granulated sugar, and cinnamon. Cut in the cold butter until crumbly. Set aside.
In a large mixing bowl, beat the butter, brown sugar, and granulated sugar together until light and fluffy, about 2–3 minutes.
Add the egg and mix until combined. Mix in the mashed banana and vanilla extract until smooth.
In a separate bowl, whisk together the flour, cornstarch, baking soda, baking powder, salt, and cinnamon.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Do not overmix.
Scoop large dough balls (about ¼ cup each), roll gently, and slightly flatten into thick discs. Place on the prepared baking sheet 2–3 inches apart.
Press the streusel topping lightly onto the tops of each cookie.
Bake for 11–13 minutes, until the edges are set and the centers remain soft.
Let cookies cool on the baking sheet for 10 minutes before transferring.
In a small bowl, whisk together powdered sugar, milk, and vanilla to make the glaze. Drizzle lightly over cooled cookies.