Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a mixing bowl, whisk together the flour, baking powder, and salt. Set aside.
In another bowl, beat the butter and sugar together until light and fluffy. Add the egg and vanilla extract, and mix until combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Add the milk and mix until smooth.
Drop teaspoon-sized balls of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 12-15 minutes, or until the edges are golden brown. Allow to cool on the baking sheet for a few minutes before transferring to a wire rack to cool completely.