Position rack in the center of the oven; preheat oven to 350°F. Spray a large baking sheet with nonstick spray. Mix flour, oats, and salt in a medium bowl; set aside.
With an electric mixer on medium, cream the shortening and both sugars until light and fluffy, about 2 minutes. Beat in the banana, then the egg white and vanilla. Turn off the mixer, remove the beaters, and stir in the flour mixture with a wooden spoon, just until incorporated. Stir in the chocolate chips.
Drop by flat tablespoons onto the baking sheet, spacing mounds about 1 1/2 inches apart. Bake until golden brown, about 15 minutes. Cool cookies on the baking sheet for 2 minutes, then cool completely on a rack. Cool the baking sheet for 5 minutes before baking additional batches. Store in an airtight container for up to 4 days.