In a large mixing bowl, beat egg yolks, salt, and sugar until thick.
In a stand mixer, beat the egg whites on high until they are stiff / glossy and hold a good peak. Do not overbeat (they should not look crumbly and dry).
Stir 1/4 of the egg whites into the egg/sugar mixture. Mix the batter until it is totally smooth. This will lighten the batter and make it easier to protect the air in the rest of the whites.
Fold in the rest of the egg whites and the flour using a large rubber spatula. Be careful to retain as much air as possible in the batter.
Line a 9 1/4 x 5 1/4 x 2 3/4 inch Large Loaf Pan with one long sheet of parchment paper. Press the paper into the pan, creasing it at the interior edges and folding the extra over the side of the pan. It is fine if it overhangs every edge by a few inches.
Pour batter into the lined pan. Bake in a preheated oven for about 40 minutes. Remove when cake tester comes out bone dry. When done, remove the pan to a rack and allow the cake to cool. The cake will shrink down as it cools.
When the cake is cool enough to cut (about an hour) use a breadknife to shave off any very dark brown patches on the sides, bottom and top. Do this as much or as little as you like. I prefer to shave most of the dark brown off. Discard the brown bits and slice the cake horizontally into slices about one third of an inch thick. You should get 6-8 slices out of the cake.
To Make The Sauce: Melt chocolate in a double boiler or the microwave. If you melt it in the microwave, stir at 30 second intervals until it is fully melted. Set aside
Mix eggs and sugar in a large sauce pan over medium heat. This can be done by hand with a whisk but a hand mixer on high is preferred.
Add the chocolate to the egg mixture and stir to combine. Continue to cook - stirring constantly- until the mixture becomes very warm to the touch. Do not overcook as the egg will cook into chunks.
Turn the heat down to low and stir in the butter and vanilla.
Clean and dry the bread pan used to bake the cake. Line it with plastic wrap so that the entire pan is lined and wrap hangs over all four edges of the pan a little bit.
Ladle some warm chocolate sauce into the bottom of the pan. The layer of the cake you put into the pan first will become the top. Place the bottom of the cake into the pan. Ladle chocolate sauce over, spreading a little so that there is a thick layer of sauce between every layer of cake. Continue to add cake and sauce until all the layers have been used. If the last layer has risen unevenly set it aside with a little sauce to eat as the cook's portion! The last layer of the cake should be covered with chocolate sauce. Reserve about 1 cup or so of the sauce to touch up your cake later as desired.
Cover the pan with plastic wrap and refrigerate for several hours or overnight until the chocolate sauce has set firm and the cake is chilled through.
Remove the plastic wrap that covers the top of the cake. Place a serving platter over the cake and flip over so the cake rests on the platter. Gently peel the rest of the plastic wrap from the cake. If the chilled sauce acts "gooey" use a paring knife to separate it from the plastic wrap. A few tablespoons of sauce will be left on the plastic wrap after it has been removed. Use the knife to gently smooth the sides and top of the cake. Add additional sauce from what you held in reserve if any bits of cake become exposed.
Serve in thin slices. Store refrigerated. Leftovers can be frozen for a few months if wrapped carefully.
Timing: 10 minutes to prep the batter. 30 minutes in the oven. 1 hour to cool the cake. 30 minutes to slice the cake, make the sauce and assemble. Several hours to chill the finished cake.