Special equipment: a large offset spatula, bench scraper and pastry bags
For the cake: Preheat the oven to 350 degrees F. Line two 18-by-13-inch baking sheets with parchment.
Add the egg whites to a CLEAN stand mixer bowl. Using the whisk attachment, whisk on high speed, adding the cream of tartar, salt and 6 to 8 generous drops of blue food coloring. Whisk until combined. Slowly add half the granulated sugar and continue to whisk on high until the meringue achieves medium-stiff peaks. Don't overmix so the meringue becomes "dry." Remove the meringue to a separate bowl; replace the mixing bowl.
Add the egg yolks and the remaining sugar to the new mixing bowl. Beat with the whisk attachment on high speed until light and doubled in volume. Add the water, oil, vanilla extract and almond extract and mix until well combined. Sift in the cake flour and mix on low until just combined. Increase the speed to high and mix for another 2 minutes. With a rubber spatula, gently fold in the egg whites.
Divide the batter between the prepared baking sheets and spread evenly using a large offset spatula. Give the baking sheets a gentle tap on the countertop to fill any gaps and remove air pockets. Bake until the cakes spring back when gently poked, 20 to 25 minutes. Ten minutes into the bake, rotate the sheets for even baking.
When the cakes come out of the oven, gently run a paring knife around their edges to release them from the baking sheets. Sprinkle both with confectioners' sugar and top each with a clean kitchen towel. Place a second baking sheet on top of each cake and then flip the cakes over. Remove the parchment. While the cakes are still warm, roll them up in the towels, starting at the short ends. Let cool about 20 minutes.
For the filling: Meanwhile, add the cream cheese and confectioners' sugar to the bowl of a stand mixer fitted with the whisk attachment. Mix on low to combine until smooth. Add the cream and vanilla extract and mix on low to combine. Increase the speed to high and whisk until stiff.
Transfer a scant 1/2 cup of the filling to a small bowl. Fold in the strawberries and a drop of red food coloring.
To assemble: Carefully unroll the cakes. Cut both cake sheets in half lengthwise. Spread the strawberry filling along the short edge of one strip of cake, covering about 2 to 3 inches. Spread the rest of the strip with plain filling. Cover the remaining 3 cake strips with plain filling.
Start rolling the cake with the strawberry filling like a jellyroll, starting at the end with the strawberry filling. Once it's rolled up, move it over to the next strip of cake, lining it up with the edge. Continue rolling up the cake into an even bigger jellyroll. Continue with the remaining strips of cake. Wrap the cake in plastic wrap and transfer it to the freezer for 30 to 40 minutes to set. In the meantime, make the buttercream.
For the Swiss meringue buttercream: In the bowl of a stand mixer, combine the egg whites, granulated sugar and salt. Place the bowl over a pan of simmering water, making sure the bottom of the bowl doesn't touch the water, and whisk until the sugar is completely melted and the mixture reaches 170 degrees F on an instant-read thermometer.
Transfer the mixing bowl to the stand mixer fitted with the whisk attachment and whisk on high until the meringue holds stiff peaks and the bowl is cool to the touch. Add the vanilla and whisk to combine. Add the butter, a tablespoon at a time, counting to 10 between each addition. When the buttercream comes together, thickening and smoothing out into a spreadable consistency, continue whisking for 2 to 3 minutes more. (Makes 5 cups.)
Place 1 1/2 cups buttercream in one bowl and add 2 drops blue food coloring; stir to combine. Add another 1 1/2 cups buttercream to a second bowl and add 2 drops of red food coloring; stir to combine. Leave the remaining buttercream plain.
To finish the cake: Stand the cake vertically on a cake stand and remove the plastic wrap. Using a large offset spatula, spread an even layer of plain buttercream onto the top of the cake. Put the remaining plain buttercream, and the red and blue buttercream into separate pastry bags (or zip-top bags) and snip off about 1/4 inch from the tip.
Pipe stripes of buttercream around the sides of the cake, alternating between the 3 colors. Use a hot bench scraper to smooth the sides of the cake (see Cook's Note).
To heat the bench scraper, keep a bowl of hot water and a clean, dry towel nearby. Dunk the scraper and dry thoroughly before using to smooth the cake. Repeat as needed.