Preheat oven to 180 C/ Gas 4. Grease a 23x33cm baking tin.
Melt vegetable fat and butter in a small pan over low heat; pour melted mixture into a mixing bowl. Stir cocoa into butter mixture until well combined, followed by sugar. Dissolve 4 teaspoons instant coffee with hot water in a small bowl; stir into cocoa mixture. Whisk eggs into batter, one at a time, beating each egg in thoroughly before adding the next. Stir flour, 2 teaspoons vanilla extract and salt into cake batter just until moist. Pour mixture into prepared baking tin.
Bake brownies in the preheated oven until a skewer inserted into the centre comes out with moist crumbs, 25 to 30 minutes. Cool completely.
Beat 115g softened butter with 1 teaspoon vanilla extract in a bowl. Add icing sugar to mixture and beat until creamy using an electric mixer on medium speed. Dissolve 1 tablespoon instant coffee with milk in a small bowl; add to icing. Increase mixer speed to high and beat until light and fluffy, 3 to 5 minutes. Spread icing on top of cooled brownies.