Special equipment: two 18-by-13-inch half-sheet pans, 2 large kitchen towels and a stand mixer with a heatproof bowl
For the cake: Arrange an oven rack in the middle of the oven and preheat the oven to 350 degrees F. Line two 18-by-13-inch half-sheet pans with parchment and coat with baking spray.
Combine the flour, baking powder and salt in a medium bowl and set aside. Whip the egg yolks with the granulated sugar in a medium bowl using an electric mixer on medium-high speed until thick ribbons form, about 5 minutes. Stir in the vanilla extract. Fold the dry ingredients into the egg yolks.
Whip the egg whites in a separate medium bowl using a hand mixer on medium-high speed until stiff peaks form, about 3 minutes. Gently fold the whites into the batter, being careful not to deflate them. Divide the batter evenly between the 2 prepared pans. Bake until lightly springy when touched, about 10 minutes.
Dust 2 clean large kitchen towels with the confectioners' sugar and flip the cakes out onto the towels with the short sides facing you. Working with 1 cake at a time, remove the parchment and roll up each cake in the towel away from you. Set aside to cool.
For the frosting: Bring a few inches of water to a boil in a saucepan that can hold the heatproof bowl of a stand mixer above the water.
Add the granulated sugar, lemon juice, salt and egg whites to the stand mixer bowl and whisk together by hand. Set the bowl above the boiling water and whisk until the mixture is warm to the touch and the sugar is completely dissolved. Transfer to the stand mixer fitted with the whisk attachment and beat at medium-high speed until cool and the whites hold stiff peaks, 10 to 15 minutes.
Continue to beat, tossing in the butter a couple of chunks at a time, making sure the pieces are incorporated before adding more. After all the butter is incorporated continue whisking on medium-high speed. The mixture will deflate and appear curdled. Continue whisking until the buttercream comes back together with a smooth and spreadable consistency, about 5 minutes. Slowly drizzle in the vanilla while beating until smooth and light, about 5 minutes more. (If the frosting is very soft or begins to break, refrigerate until set but still spreadable and then beat until light before using.)
Transfer half of the frosting to another bowl and set aside. Add the food coloring to the remaining half of the frosting and beat to incorporate.
To assemble: Unroll the cooled cake layers so they lay flat. Transfer the cakes to a cutting board and evenly spread each with a 1/8-inch-thick layer of the colored frosting. Use a serrated knife to divide each cake in half lengthwise, creating 2 long strips of cake (4 total).
Take one strip of frosted cake and roll it into a tight jelly roll. Add another strip of cake, matching the ends and coiling it around the center roll as you would a cinnamon roll; repeat with the remaining cake strips, trimming the ends to fit as necessary. Chill in the refrigerator for 30 minutes.
Trim the ends of the cake so they are flat and transfer to a cake board or platter, standing the cake up on one end and securing it to the surface with a dollop of the reserved white frosting.
Spread a thin layer of the white frosting over the top and sides of the cake for a crumb coat. Chill the cake in the refrigerator for 30 minutes to set the coat.
Frost the cake with the remaining white frosting, using an offset spatula to create swirls if desired. Decorate the top edge of the cake with sprinkles as desired.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.) You can make the frostings and store them in airtight containers in the refrigerator up to 3 days in advance. Allow them to come to room temperature and mix on medium-high until smooth before using.
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