Preheat oven to 350°F. Grease a 9 x 9-inch baking pan and line it with parchment paper or foil, leaving an overhang for easy removal; grease the lining.
Whisk together 1 ½ cups flour, ½ teaspoon baking powder, and ½ teaspoon salt in a bowl. Set aside.
In a heatproof bowl, gently melt ⅔ cup butter (10 tablespoons plus 2 teaspoons) and 3 (1-ounce) squares unsweetened chocolate, stirring frequently until smooth. Remove from heat and let cool slightly.
In a large bowl, combine 2 cups sugar and 3 large eggs. Beat or whisk until the mixture is lighter in color and slightly thickened.
Stir the melted butter-and-chocolate mixture into the sugar-and-egg mixture until combined.
Stir in ¼ cup Kahlua until evenly mixed.
Add the flour mixture to the wet ingredients and fold just until no streaks of flour remain; do not overmix.
Scrape the batter into the prepared pan and smooth the top. Bake for 30 minutes, or until the top springs back when touched lightly in the center and the edges begin to pull away from the pan. (If using an 8 x 8-inch pan, bake about 35 minutes.)
Cool the brownies completely in the pan on a wire rack. Once cooled, brush the top with the remaining 1 tablespoon Kahlua and allow it to absorb before removing from the pan and cutting.