Preheat oven to 350°F (175°C).
Line the bottom and sides of a 9-inch square baking dish with aluminum foil, allowing 2 to 3 inches to extend over the sides. Lightly grease the foil.
In a large microwave-safe bowl, combine ¾ cup semi-sweet chocolate chips and ¾ cup butter. Microwave on HIGH for 1½ to 2 minutes, stirring every 30 seconds, until melted and smooth.
Whisk 2 cups sugar into the melted chocolate and butter mixture until well combined.
Add 4 large eggs one at a time, whisking just until blended after each addition.
Gently stir in 1½ cups all-purpose flour, 1 (1-oz) bottle green liquid food coloring, 1½ tsp baking powder, 1 tsp vanilla extract, and ⅛ tsp salt until just combined.
Pour the batter into the prepared pan. Bake for 44 to 48 minutes, or until a wooden pick inserted in the center comes out with a few moist crumbs.
Cool the brownies completely in the pan on a wire rack, about 2 hours.
In a mixing bowl, beat 1 (8-oz) package softened cream cheese and 3 Tbsp softened butter on medium speed until creamy.
Gradually add 1½ cups powdered sugar and ⅛ tsp salt, beating until blended. Stir in 1 tsp vanilla extract.
Lift the brownies from the pan using the foil as handles. Carefully remove the foil.
Spread the cream cheese frosting evenly over the top of the cooled brownies.
Cut into 16 squares.