Heat the oven to 350°F (177°C). Line a baking sheet with parchment paper or a silicone mat and set aside.
In a medium bowl, mash the ripe bananas with a fork until mostly smooth. Stir in the peanut butter and vanilla until the mixture is uniform and slightly glossy.
In a separate larger bowl, whisk together the quick oats, cinnamon, salt and ground flaxseed so the dry flavors are evenly distributed.
Pour the banana-peanut butter mixture into the dry ingredients and fold gently with a spatula until everything is combined. Fold in the chocolate chips last, keeping some on top if you like a chocolatey finish.
Let the batter sit 8 to 10 minutes at room temperature so the oats absorb moisture and the mixture firms up slightly — this helps the cookies hold their shape while baking.
Use a 2-tablespoon cookie scoop or two spoons to portion the dough onto the prepared sheet, spacing them about 1 1/2 inches apart. Press each mound down gently with the back of a spoon to form a round cookie about 1/2 inch thick.
Bake on the middle rack for 12 to 16 minutes, until the edges look set and the bottoms are lightly golden. Rotate the pan halfway through baking for even color.
Allow the cookies to cool on the baking sheet for 4 to 6 minutes, then transfer to a wire rack to finish cooling. The centers will firm up as they cool.
Store cooled cookies in an airtight container at room temperature for up to 3 days, or refrigerate for up to a week. They also freeze well for longer storage.