Position a rack in the middle of the oven. Preheat the oven to 350° F. Line two baking sheets with parchment paper.

Put the chocolate in a heatproof container or the top of a double boiler, and place it over, but not touching, a saucepan of barely simmering water (or the bottom of the double boiler). Stir the chocolate until it is melted and smooth. Remove from the water and set aside.

Sift the flour, baking powder, and salt into a medium bowl and set aside. In a large bowl, using an electric mixer on medium speed, beat the butter, brown sugar, and vanilla until smoothly blended, about 1 minute. Stop the mixer and scrape the sides of the bowl as needed during mixing. Add the eggs and mix until blended, about 1 minute. On low speed, add the flour mixture, mixing just until it is incorporated.

Drop heaping tablespoons (about 3 level tablespoons each) of dough 3 inches apart onto the prepared baking sheets. Drizzle about ½ teaspoon of the melted chocolate over the top of each cookie. Using a small sharp knife, gently swirl the chocolate once or twice through the cookie to marbleize it.

Bake the cookies one sheet at a time until the tops feel soft but set and the edges are lightly browned, about 11 minutes. Cool the cookies on the baking sheets for 10 minutes, then use a wide metal spatula to transfer them to a wire rack to cool completely.

The cookies can be stored in a tightly covered container at room temperature for up to 4 days.