The proportions in this recipe will make any of:
Preheat the oven: use to 400° F for sheets, 375°F for layers.
Cut out parchment paper to fit the bottom of the pan. Butter the bottom and sides of the pan; insert and smooth out the parchment. Butter the top of the parchment, then slightly flour all the buttered surfaces, and set aside.
Gently melt the butter in a small heavy pot. You can add the vanilla to this; set it aside, but don't let it cool.
Bring the water to a boil in a heavy pot, and whisk in the sifted cocoa. If it's too stiff to mix, add a little more water, but not too much. Cover and set this aside; you don't want it to cool too much, nor dry out.
Mix the eggs and sugar together (whisk just lightly to mix) and warm in a warm-water bath up to a hot bath temperature (about 100 to110° F). Don't cook the eggs! This is easier if they start at room temperature, then you put the bowl in a water bath that is only hot-bath temperature; then you can just stir occasionally while you prepare other things.
Whip the egg and sugar mixture mixture at high speed until it reaches what is called the "au roubin" stage. With well-warmed eggs and a big stand mixer, this takes only about 5 minutes; it may take 10-15 minutes with colder eggs and a less powerful mixer. The "au roubin" stage is reached when, if you scoop out some batter and let a trail of it pour back into the bowl, it leaves a trail (ribbon) which does not go away.
Scrape some of the batter, about a cupful, into the pot or bowl with the moistened cocoa. Whisk until smooth.
Do the same with the butter. We use a hand mixer at high speed; the batter loses very little volume as a result, and is then easy to fold into the main batch of batter later without deflating it.
Sift the flour over the main batch of batter and fold it in, in three or four batches. Scrape the cocoa/batter mixture in, and fold it in. Scrape the butter/batter mixture in, and fold it in. When folding in a sequence of additions like this, proceed to the next addition before the current one is thoroughly blended in; this minimizes the amount you work, and deflate, the batter. Scrape the batter into the prepared pan(s), and smooth it out. Put into the oven.
This cake in a single sheet will cook in about 13-15 minutes; spread thinly in two sheets, somewhat less; and in deeper pans, 20 to 30 minutes depending on the exact thickness and your oven conditions. It is done when the sides shrink a bit from the pan, and the center is springy. If lightly pressing a finger on it leaves a depression that doesn't go away, it needs to cook a little longer. Beware that the thin sheets can cook dry and crisp at the edges if you oven hot spots, or just leave it in too long. Remove from the oven. Loosen the sides of the cake from the pan with a knife (usually you will only need to press it away from the side to loosen the lower corner, rather than cut it).

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