Preheat oven to 350 degrees F. Grease an 8 by 8-inch baking dish
In a small saucepan add the butter and semi sweet chocolate chips. Over medium low heat melt the two ingredients stirring often. Once melted cool slightly.
In a medium bowl combine the melted chocolate and sugar and mix with an electric mixer. Beat in the eggs one at a time. Mix in the vanilla.
Add the flour and mix in until blended. Fold in the 1/2 cup milk chocolate chips. Pour the brownie batter into the prepared baking pan. Place in the preheated oven. Bake for 25 to 30 minutes. Remove from the oven and cool completely on a cooling rack.
While the brownies are cooling whip up the frosting. In a small sauce pan combine the milk and the 2 1/2 tablespoons of flour. Over medium low heat cook until the ingredients thicken stirring during this process.
Remove from the heat and transfer the mixture to a medium mixing bowl. Once the mixture is cooled add the butter, sugar, and cocoa, beat with an electric mixer until fluffy about 2 to 5 minutes. Add the vanilla and mix in.
Frost the cooled brownies and enjoy.