Preheat oven to 350F and line a baking sheet with parchment paper.
In a medium bowl, whisk together flour, baking powder, baking soda, salt and cinnamon.
In a large bowl, cream together the butter and sugar. Beat in the egg, followed by the carrots and vanilla. Using your hands, stir in the flour mixture and the oats until just combined, adding the milk 1 tablespoon at a time until fully combined. This will be really really sticky, don't worry!
Drop tablespoonfuls of the dough onto the prepared baking sheet, flattening each cookie slightly. Bake for about 8-10 minutes.
Let cool on sheet for 3 or 4 minutes, then transfer to a wire rack to cool completely.