Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking pan.
Blend cake mix, 1 cup milk, 1/2 cup butter, eggs, and 2 tablespoons vanilla extract in a blender until batter is smooth, about 3 minutes. Pour batter into the prepared pan.
Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, about 30 minutes.
Heat 1/2 cup butter, light brown sugar, and dark brown sugar in a saucepan; bring to a boil. Add 1/4 cup milk and 1 teaspoon vanilla extract and return to a boil. Remove saucepan from heat and spoon confectioners' sugar, about 2 tablespoons at a time, into mixture using a wooden spoon, stirring after each addition until frosting is smooth. Spread frosting over cake.