Cake: set an oven rack in the middle of the oven; preheat oven to 350°; butter three 8-inch round cake pans; line them with parchment or wax paper cut to fit; then butter the paper; dust all over with fine dry bread crumbs, invert the pans over paper and tap gently to shake out excess; set aside.
Sift together the flour, baking powder, salt, ginger, and nutmeg; set aside.
In a big bowl of an electric mixer, beat the sugar and oil just to mix.
Add in egg yolks; beat to mix.
On low speed, add the sifted dry ingredients in three additions alternating with the boiling water and vanilla in two additions; remove the bowl from the mixer and set aside.
Note: to prepare the potatoes—peel them and then grate them on the fine grater of a food processor or with a hand-held grater set over a piece of foil; use the side of the grater that has small, round—not diamond-shaped—openings; measure 2 cups tightly packed.
Stir the potatoes and then the pecans into the batter.
In a small bowl of an electric mixer (with clean beaters), beat the whites until they hold a straight shape when the beaters are raised, but are not stiff or dry.
Without being too thorough, fold about one-third of the whites into the batter, then fold in the remaining whites, handling as little as necessary until just incorporated.
Divide the batter among the prepared pans and smooth the tops.
Bake the three pans on the same oven rack for 30-35 minutes, until the cakes barely begin to come away from the sides of the pan (these layers might not spring back when pressed with a fingertip, even though they will be done).
Cool in the pans for 2-3 minutes; then cover each pan with a rack, turn the pan and rack over, remove the pan and paper lining, cover with another rack, and turn over again, leaving the layers right side up to cool.
When you are ready to ice the cake, place four 12x4 inch strips of parchment or wax paper in a square pattern around the sides of a large cake plate; place one layer on the plate.
Stir the apricot preserves in a small saucepan over medium heat to melt.
Then press through a strainer; with a teaspoon or pastry brush, spread the first cake layer with 1/3 of the preserves; reserve the remaining preserves.
Icing: add the sugar, cream of tartar, and water in a 6-cup saucepan; with a wooden spoon stir over medium heat until the mixture begins to boil.
Cover airtight and let boil 3 minutes (this keeps the steam in the pot and dissolves any sugar crystals that cling to the sides).
Uncover and insert a candy thermometer; increase heat to high and let boil, without stirring, until the thermometer registers 242°.
Shortly before the sugar syrup is done (or when the thermometer registers about 236°--soft ball stage), add the salt to the egg whites in the large bowl of an electric mixer and beat until the whites are stiff (if the sugar syrup is not ready, turn the beater to lowest speed and let it beat slowly until the syrup is ready; or you can let the whites stand, but no longer than necessary).
When the syrup reaches 242°, turn the mixer to high speed and gradually add the syrup in a thin stream (it may be easiest if you pour the syrup into a pitcher and add it from the pitcher).
Then continue to beat at high speed, scraping the bowl occasionally with a rubber spatula, for about 5 minutes or until the icing is quite thick and stiff (if necessary, beat some more; the icing may still be warm when it is used).
Add the vanilla and almond extract a minute or two before the icing is stiff enough.
Spread the first cake layer with icing about ½ inch thick.
Place the second layer over it and spread with 1/3 of melted apricot preserves, and then cover it with a layer of icing about ½ inch thick.
Cover with the third cake layer and the remaining preserves.
Now, it is best to ice the sides first; use a long, narrow metal spatula to ice the sides, thinly at first, and then build it up until it is about ½ inch thick, or thicker.
Smooth the sides; use the remaining icing on the top; spread it smooth.
After the sides and the top are smooth and even, then, with the back of a teaspoon or with the spatula, form swirls and peaks evenly on the top of the cake.
To coat the sides with the coconut, first spread out the coconut the length of a foil or wax paper next to the cake plate.
Take a handful of the coconut in the palm of your hand and turn your hand to place the coconut on the sides of the cake, starting at the top and working your way down.
When much of the coconut falls onto the plate, remove it with your fingers and replace it either on the cake or on the pile of coconut on the paper.
Then, use a long, narrow metal spatula to pick up the coconut that has fallen to the plate and return it onto the sides of the cake.
Finally, fold the paper strips around the bottom up against the cake, and the coconut that has fallen to the strip will stick to the base of the cake; last, sprinkle all the remaining coconut over the top.
Remove the paper strips by pulling each one slowly and gently toward a narrow end.
Since the cake is so high, use a long-bladed knife to cut it and dinner plates to serve it on.