Preheat the oven to 350°F (176°C) and line baking sheets with parchment paper or silicone baking mats. You'll need 3 to 4 baking sheets, but if you don't have that many, just reuse what you have.
In a medium bowl, whisk together the flour, baking soda, baking powder and salt.
In a large bowl, cream the butter, sugars and pudding mix on high speed until light and fluffy. Beat in the eggs and yolk, one at a time, until incorporated. Mix in the mashed banana and vanilla. The batter will be a little lumpy.
Gradually add in the flour mixture on low speed, until just combined. Fold in the chocolate chips by hand.
Cover and chill the dough in the refrigerator for 1 hour.
Scoop 12 balls of dough onto a baking sheet. Each ball should be about 1.5 inches. Press the top of each cookie dough ball with the back of a spoon to slightly flatten it.
Bake for 10 to 12 minutes, or until the edges are lightly browned. Let the cookies rest on the pan for 5 minutes before removing them to a wire cooling rack.
Store in an airtight container. These cookies also freeze well.