Preheat oven to 350 degrees F. Grease a 9 by 13-inch baking pan.
Whisk together flour, baking powder, and salt. Set aside.
Using an electric mixer on medium speed, beat the butter, sugar, and brown sugar until light and fluffy. Add the eggs, one at a time, mixing well after each addition. Mix in the vanilla.
Reduce mixer speed to low. Gradually add the flour mixture, mixing just until combined.
Drop slightly more than half the batter by tablespoonfuls into the prepared pan, leaving gaps between the portions.
Add the cooled chocolate to the remaining dough, and mix just until combined. Drop the chocolate dough by tablespoonfuls in between the vanilla dough in the pan. The dough will spread as it bakes, but you can gently spread and smooth the doughs if you like.
Bake 30 to 35 minutes, or until the edges are browned and a pick inserted into the center comes out clean.
Cool in the pan for at least 15 minutes before cutting into bars.