Sprinkle 1/2 cup sugar in a heavy saucepan; place over medium heat. Cook, stirring constantly, until sugar melts and syrup is light golden brown.The sugar will start to clump, this is normal, just keep stirring.
Combine reamaining 2 1/2 cups sugar, milk, beaten egg, and salt, stirring well, stir in the butter. Stir mixture into carmelized sugar, the mixture will lump again, just keep stirring.
Cook over medium heat stirring frequently, until a candy thermometer registers at 230F, this takes about 15-20 minutes, but don't rely on the time! Check the thermometer!)Cool mixture 5 minutes.
Beat mixture with a WOODEN spoon to almost spreading consistency. The mixture will cool and thicken quickly at this point. Spread on a cooled cake.