Place a rack in the upper third of the oven and preheat to 350 degrees F. Grease and flour an 8-inch square baking dish and set aside.
In a medium bowl, whisk together flour, baking powder, and salt. Set aside.
Bring about 2 inches of water to a boil in a medium saucepan. In a medium, heatproof bowl, add butter and unsweetened chocolate. Place the bowl over, but not touching, the simmering water and allow to melt. Stir to incorporate. Once melted, remove the bowl from the simmering pot. Whisk in brown sugar and granulated sugar. Whisk in eggs, yolk, and vanilla. Add the flour mixture all at once to the chocolate mixture. Fold together with a spatula until well incorporated. Fold in the chocolate chips.
Pour batter into prepared pan and bake for 25 to 30 minutes. Bake until a skewer inserted in the center of the brownie comes out clean. Remove from the oven and cool completely before frosting.
Frost with a half recipe of The Best Chocolate Buttercream Frosting (see below) when cooled.Brownies last, well wrapped, at room temperature for up to five days.