Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.
Stir milk and vinegar together in a small bowl to make sour milk. Sift flour, baking powder, baking soda, and salt together in a separate bowl.
Beat white sugar and 5 tablespoons butter together in a large bowl with an electric mixer until light and fluffy. Add 2 eggs; beat until smooth. Beat in portions of sour milk and flour mixture alternately until smooth batter forms.
Melt chocolate in top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching, 1 to 3 minutes. Stir melted chocolate and vanilla extract into batter. Pour batter into prepared baking dish.
Bake in preheated oven until a toothpick inserted into the center comes out clean, about 25 minutes. Cool in the baking dish for 10 minutes before removing to cool completely on a wire rack.
Beat 1/2 cup butter in a large bowl with an electric mixer until smooth and creamy. Beat in egg yolk; gradually beat in confectioners' sugar until light and fluffy. Mix in lemon juice until smooth frosting forms. Frost cake with frosting when cake has completely cooled.