Preheat the oven to 350 degrees F (175 degrees C).
Combine flour, baking soda, baking powder, and salt in a small bowl.
Beat eggs lightly in a large bowl. Add brown sugar, white sugar, shortening, and peanut butter. Mix until well blended and fluffy. Add 1/2 of the flour mixture; mix well. Repeat with remaining flour mixture; cookie dough should appear slightly dry and crumbly. Add bananas and mix well.
Roll dough into small walnut-sized balls. Place on baking sheets. Press criss-cross patterns into each cookie using a fork, dipping it into some flour to prevent sticking.
Bake in the preheated oven until edges are golden, about 10 minutes. Let cool completely, at least 20 minutes. Apply some chocolate syrup to the back of a tablespoon and drizzle it over each cookie.
