Grease a jelly-roll pan with plenty of softened butter, then line with wax paper or parchment. Grease again.
In the bowl of a stand mixer fitted with the whisk attachment, beat the egg yolks and 4 tablespoons (¼ cup) of the sugar until thick. Beat in the cocoa powder, confectioners' sugar, oil, salt and vanilla, scraping side of bowl as you go. Remove bowl from the stand.
Put the bowl with the egg whites on the mixer stand and with the whisk attachment, beat on medium until they are frothy. Increase speed and beat until soft peaks form, then add the 2 tablespoons sugar gradually. As the egg white start to stiffen, beat in the vinegar. Just beat until stiff peaks form, don't let the mixture get too dry.
With a heavy scraper, fold together the egg whites and the chocolate egg yolk mixture.
Pour the batter into the pan and spread it evenly (I do it by tilting the pan). Bake on the center rack for 15 minutes, or until it is puffed and just set.
Remove from the oven and let cool completely. You can cover it loosely with another sheet of wax paper if you like. While it's cooling, carefully loosen the cake from the edges of the wax paper with the tip of a knife.
Filling: When cake is cool, prepare the filling. In a chilled mixing bowl beat the cream until soft peaks start to form, then gradually beat in the sugar. Beat in the Amaretto.
When cake is cool, mix together the extra confectioners' sugar and cocoa and sift over the top. Cover the cake with a sheet of lightly greased wax paper or parchment, then lay something large like a big cutting board or baking sheet over the wax paper. Invert the cake so that it sits on the greased wax paper. Peel off the wax paper that was previously underneath it. Spread the filling over the roll evenly, leaving an inch border on the sides to give the filling room to squish out.
Starting at the short side, roll up the cake jelly-roll style, slightly lifting the wax paper to help guide the cake into a roll. It will most likely crack slightly, but that is okay. Bring the waxed paper up and around the roll and put the whole thing in the freezer. Freeze until very firm and easier to handle. Meanwhile, make the icing.
For the icing, beat the butter until creamy. Add the cocoa powder and stir or beat on low until blended, then beat in the powdered sugar. Add milk one tablespoon at a time, beating until smooth, then beat in the Amaretto.
Spread icing all over the cake. At this point you can put it on a nice serving plate and decorate or you can wrap the iced cake in plastic wrap and freeze until the next day. When frozen, set the whole thing on a nice serving plate, allow it to thaw somewhat, and decorate with sprinkles. Store in the refrigerator until serving time.