Baking Preparations: Position the rack in the lower third of the oven; preheat oven to 350°F.

Using a paper towel, grease the bottom and sides of the pan with solid shortening. Dust generously with all-purpose flour, shake to distribute, and tap out the excess.

Place a cooling rack on a counter that is not likely to be in the way. The chocolate cake will be removed to this rack after baking, then glazed immediately and left to sit.

Ingredient Preparations: Melt the 4 ounces of butter in a small saucepan. Remove from heat; cool for 5 minutes; then add the water and oil.

Break the eggs into a bowl; whisk to combine. Add the buttermilk and vanilla and stir to combine.

Pour the flour, granulated sugar, cocoa, baking soda, and salt in that order into a triple sifter over the bowl of a heavy-duty mixer. Sift into the bowl.

Attach the bowl to the mixer and add the butter and water mixture.

Making the Cake: With the flat beater (paddle), blend the ingredients on medium-low speed (#3) for about 30 to 45 seconds. Add the buttermilk mixture; resume the same speed and blend just until smooth (about 30 to 45 seconds). Remove the flat beater and bowl from the mixer. Tap the beater against the edge of the bowl to remove excess.

Baking the Cake: Pour the batter into the prepared pan, beginning around the edge of the pan and then into the center. With the rubber spatula, manipulate the batter from the center so the batter is higher along the sides (this assures a more level surface after baking). Bake for 20 to 22 minutes, or until the cake springs back slightly when lightly touched in the center and the sides begin to contract from the pan.

Making the Glaze: While the cake is baking, begin the glaze preparation. Place a sieve over a 1½-quart mixing bowl; set nearby.

Pour the powered sugar and cocoa in that order into a triple sifter, and sift into a 3-quart mixing bowl.

Pour the milk into a small saucepan, add the 4 ounces of butter, and melt over low heat. Pour the hot milk mixture into the powdered sugar mixture and stir to combine. Add the vanilla and blend. Pour the mixture through the sieve over the bowl, and push it through with the aid of a rubber spatula.

Finishing the Cake: When the cake has finished baking, remove it to the cooling rack. Immediately pour the glaze (its consistency is thicker than heavy cream, but runny) over the hot cake from the top of the pan’s short side down almost to its other short end. Spread the glaze with the spatula over the surface. Do not worry if some glaze touches the pan’s rim; this will be removed later. Spread the glaze just to cover; then stop. (The heat from the cake will keep the glaze liquid enough to spread, but too much manipulation of the glaze will dull its finish.)

Cooling and Cutting the Cake: Do not touch or move the cake for at least 4 hours or the glaze will crack. (Waiting overnight is fine. Don’t cover or refrigerate it.) The glaze’s surface will set as the cake cools.

To cut the cake into neat individual squares, you need a sharp small paring knife, a small glass of hot water, a paper towel, and an 18-inch ruler.

First, free the glaze sticking to the rim of the pan. Dip the blade of the knife into the hot water, wipe it with the paper towel, and then trace the perimeter of one side of the pan to release just the glaze, not the cake. Repeat this procedure along the sides. (Now the baking pan may be picked up without the glaze’s cracking.) Remove the cooling rack from under the pan so it does not tilt when you cut the squares.

Place the ruler next to one of the longest sides (15 inches). With the tip of the knife blade mark 1/16-inch-long notches in the glaze every 1½ inches (or larger) along the entire length. Repeat on the opposite side, so that the notches line up directly across from each other. Repeat this procedure on the short sides of the pan, too.

With the paring knife, dipped into hot water and wiped, score just the glaze (not cutting through the cake) in a straight line from one notch to the other, using the ruler as a guide. When you are finished, the pattern for the squares has been formed.

Now cut through the scored lines, heating and cleaning the paring knife each time. Decorate each square, if desired.