Preheat the oven to 350°F (180°C). Line a large cookie sheet with parchment paper.
Place flour in a large bowl.
In a small bowl, whisk eggs, peanut butter and banana until blended. Stir in water.
Pour egg mixture into the flour, stirring with a fork until the mixture starts to come together. With the mixture still in the bowl, use your fingers to work the mixture into a ball. The dough should be a bit stiff but if it seems quite dry and won’t form into a ball, add a little more water, 1 tablespoon (15 mL) at a time until it comes together.
Place the dough on a lightly floured work surface. Knead it gently until it smooths out and feels combined. Using a rolling pin, roll out the dough to about 3/16 in (5 mm) thick. Cut into shapes using cookie cutters, cutting the shapes as close to each other as possible to avoid having to reroll a lot of dough a second time. Using a 3 inch (8 cm) cookie cutter will yield about 40 cookies. Alternately, use a knife or pizza cutter to cut the dough into strips, squares or random shapes.
Place the shapes on the cookie sheet. (Sometimes it’s helpful to slip a knife or thin-edged pancake turner underneath the dough shapes to slide the cookies off the work surface and preserve their shapes.)
Bake in the preheated oven until the cookies are baked through, feel very firm to the touch, and are starting to brown on the bottom, 20 to 25 minutes.
Meanwhile, if you used cookie cutters, gather any scraps of dough into a ball and roll out again. Recut using the cookie cutters, or cut into strips or squares with a knife. Bake these remaining pieces; if they are smaller than the first batch of cut-outs, they make take a few minutes less to bake.
When the cookies are removed from the oven, transfer them from the cookie sheet onto a wire rack to cool.
When cool, place in an airtight container and store in the refrigerator for up to two weeks.