Preheat oven to 350°F. Spray a 9×9 glass baking pan with nonstick cooking spray and set aside.
In a small saucepan over medium-low heat, melt 1/2 cup butter, 3/4 cup semi-sweet chocolate chips, and 3 tablespoons unsweetened cocoa powder, stirring until smooth. Remove from heat.
Stir 1 cup granulated sugar into the melted chocolate mixture until combined.
Add 2 eggs, one at a time, stirring after each until incorporated. Stir in 3/4 teaspoon vanilla extract.
In a separate bowl, whisk together 1 cup all-purpose flour, 1/4 teaspoon baking powder, and 1/2 teaspoon salt.
Gently fold the dry ingredients into the chocolate mixture until just incorporated—do not overmix.
Pour the batter into the prepared pan, smooth the top, and bake at 350°F for 20–30 minutes, or until the brownies are just set and a toothpick inserted in the center comes out with a few moist crumbs. Remove from oven and cool in the pan on a wire rack until completely cool.
For the frosting, stir together 3 tablespoons softened butter, 3 tablespoons cookie butter or peanut butter, 1/2 teaspoon vanilla extract, and 1 cup powdered sugar until a thick paste forms.
Stir in 1 tablespoon milk a little at a time until the frosting reaches a spreadable consistency, then mix vigorously to smooth any lumps.
Spread the frosting evenly over the cooled brownies, cut into bars, and serve.