Preheat oven to 375 degrees F (190 degrees C) with the rack in the middle position. Grease baking sheets or line with parchment paper. Sift together the flour, baking soda, and salt.
Break the chocolate squares into chunks, place in a microwave-safe bowl and microwave on High until the chocolate melts, about 90 seconds. Stir the chocolate until smooth and set aside to cool.
In a large bowl, beat 1/2 cup butter, brown sugar, and white sugar until light and fluffy; pour in the egg and beat until smooth. Mix in the red food coloring and chocolate, scraping the bowl down regularly, until evenly blended, about 30 seconds. Add half of the sifted dry ingredients, stirring until well incorporated. Beat in the sour cream and mix in the remaining dry ingredients. Fold in the chocolate chips. Drop spoonfuls of the dough 2 inches apart onto prepared baking sheets.
Bake one sheet at a time in the preheated oven until they spring back when pressed, about 9 minutes. Cool in the pans for 5 minutes before removing to cool completely on a wire rack.
For the cream cheese frosting, whip 1/4 cup butter, cream cheese, and vanilla until smooth. Blend in the powdered sugar in half cup portions until the frosting reaches the desired consistency.