Preheat oven to 400°F - 200°C Cover a baking sheets with a cooking paper.
Cream together butter, honey, mushed banana and egg white. Sift the flour with the vanilla extract and stir in the butter cream. Whisk to combine. Drop by teaspoonfuls onto the covered baking sheets. Brush with milk. You can sprinkle some cinnamon or sugar substitute if you like. Bake about 15 minutes or until lightly brown.