Position a rack in the upper third of the oven. Preheat the oven to 375ºF (190ºC). Line two baking sheets with parchment paper or silicone baking mats.
Sift together the flour, baking powder, and salt into a medium bowl and set aside.Combine the vanilla, milk, and lemon juice in a small bowl or a measuring cup and set aside.In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl using a handheld mixer, beat the butter on high speed for about 1 minute, until light and airy.Gradually mix in the granulated sugar. Add the egg and beat until incorporated, then stop to scrape down the bowl.
With the mixer on medium-low speed, alternately with the flour and milk, mixtures in three additions, starting and ending with the flour mixture and mixing until just combined after each addition; stop to scrape down the bowl as often as necessary.
Use a cookie scoop or a tablespoon to drop 1-tablespoon mounds of dough onto a prepared baking sheet, spacing them about 1½ inches (4 centimeters) apart. Bake in the upper third of the oven for 8 to 10 minutes, until the cookies are firm to the touch and fragrant. Repeat with the remaining batter. Let the cookies cool on the baking sheet for 10 minutes, then transfer them to a wire rack set over a sheet of parchment paper. Turn them over—the flat underside will become the top side of the cookie for easy glazing—then cool completely.
Make the glazes: Combine the confectioners’ sugar, ¼ cup (60 milliliters) of the milk, and the vanilla in a small bowl and stir until smooth. Holding a cookie in one hand over the bowl of glaze, use a small spoon to scoop up the glaze and pour it over half the cookie, then, with the back of the spoon, move the glaze around to completely coat half the cookie. Push any excess glaze off the edges so as to fully cover the half. Return the half-glazed cookie to the rack.Repeat with the remaining cookies.
Add the cocoa powder to the glaze remaining in the bowl, then add a teaspoon or two of additional milk, just enough to loosen it—you want it opaque enough to cover the cookie, but thin enough to be workable. Mix until smooth.
Glaze the unglazed half of each cookie as you did with the white glaze. Return them to the rack to set for 30 minutes before eating them or packing them up.
The cookies will keep in an airtight container at room temperature for up to 3 days.