For the cake: Heat the oven to 375 degrees F. Line two 9 1/2 x 14-inch jelly-roll pans with parchment paper, leaving 1 inch of paper hanging over on all sides (this will help you lift the cakes out of the pan after baking).
In the bowl of a stand mixer fitted with the whisk attachment, whip the yolks and 1 cup of the sugar on medium speed until the mixture is thick and fluffy, 3 to 4 minutes. Add the vanilla and mix for 1 minute. Scrape into a medium mixing bowl and set aside.
Wash the stand mixer bowl and the whisk attachment very well (even the smallest amount of yolk will prevent the whites from whipping properly). Put the whites and salt in the mixer bowl whisk at medium speed until the whites become frothy. Slowly add the remaining 1/2 cup plus 2 tablespoons sugar. Increase the speed to high and whip until the whites become glossy and almost firm (the whites should not become dry). Remove the bowl from the mixer.
Sift the cocoa over the egg yolk mixture and gently fold in one-third of the whites (this will help to prevent overmixing). Add the remaining whites and carefully fold until there are no visible streaks of white.
Divide the batter between the two pans. Smooth the batter evenly with an offset spatula. Bake the cakes just until they have risen and feel like soft marshmallows, 8 to 10 minutes. Transfer the pans to wire racks and let the cakes cool to room temperature. Chill the cakes until cold, about 30 minutes. While the cakes are cooling, prepare the buttercream.
For the peanut buttercream: In the bowl of a stand mixer fitted with the paddle attachment, cream the butter and peanut butter on medium speed until fluffy, about 2 minutes. Turn the speed to low, add the confectioners' sugar, salt and vanilla and mix for another 2 minutes. Add the cream, turn the speed to high, and beat for 1 minute. Cover and set aside.
When cooled, remove the cakes from the refrigerator. Gently lift one cake from the pan, using the overhanging parchment as handles, and place it on a cutting board. Carefully separate the paper from the cake. Lay a piece of parchment on top of the cake and quickly flip the cake over. Remove the parchment and replace with a clean sheet of parchment. With a rolling pin, firmly roll over the cake to even it out and thin it out.
Trim each cake to a 12-by-9-inch rectangle. Place in the freezer for 15 to 20 minutes; this firms it up, and if the cake is stuck to the parchment this allows you to flip one more time to release the paper. Repeat with the remaining cake.
Spread a thin layer of marshmallow cream over the surface of both cakes. Divide the buttercream between the cakes and spread it evenly over the jelly, leaving a border at the top of one long side. (The rolls look best when the cake layer and buttercream layer are the same thickness.)
Begin rolling up the cakes, one at a time, starting at a long side. The first roll is the most important-keep it tight. Continue rolling toward the top. It's best to go slowly and carefully. When you completed rolling a cake into a log, place it in the center of a new sheet of parchment. Take the paper above the roll and cover the roll with a little overlap. Now take a flat-edge utensil, like a bench scraper or a long spatula, and force it underneath the roll and push tight. This helps fill in any air pockets in the roll. Chill the rolls 15 to 20 minutes or overnight before cutting them; the colder, the better.
For the chocolate shell: Put the chopped chocolate in a heatproof bowl and set it over a pan with 3 inches of simmering water. When the chocolate is completely melted, stir in the butter and oil. Set aside in a warm place.
Remove the paper from the rolls. With a sharp knife, cut each roll into 6 even pieces; each will be about 2 inches wide. Insert a popsicle stick into each piece.
Place a screen over a sheet pan to allow excess chocolate to drip off. Working with one roll at a time, dip it into the bowl of chocolate shell, turning it to coat completely. Tap off excess chocolate before setting it on the screen. Continue until each roll is completely coated. While on the screen, move each roll slightly so it doesn't stick to the screen while drying.
Once dry, carefully lift off and arrange the pops on a sheet pan. Freeze until they're hard enough to wrap. Wrap securely and freeze for up to 3 weeks. They're a great snack for any time of day.
The chocolate shell mixture can be stored in the refrigerator for up to 2 weeks and gently reheated. It's great poured over ice cream or sorbet.