Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 10x16-inch baking dish.
Beat eggs, chocolate syrup, flour, white sugar, 1/2 cup softened butter, vanilla extract, and salt together in a bowl until smooth; pour into prepared dish.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 25 minutes. Cool in the pan completely, at least 30 minutes.
Beat confectioners' sugar, 1/2 cup melted butter, milk, and peppermint extract together in a bowl until smooth; spread over the cooled brownies. Wait for frosting to set before cutting to serve, about 10 minutes.