For the cake: Preheat the oven to 350°F. Grease a 13-by-9-by-2-in pan with nonstick cooking spray or softened butter and line with parchment paper.
Add the flour, cocoa powder, brown sugar, granulated sugar, baking powder, baking soda, and salt to the bowl of a stand mixer fitted with the paddle attachment; mix on low speed until well combined.
In a small bowl, or in a large 4-cup liquid measuring cup (my preference), add the egg, egg yolks, vanilla, oil, and sour cream and whisk until combined. With the mixer on medium-low speed, slowly pour the wet ingredients into the dry ingredients. Beat until incorporated, scraping down the bowl with a rubber spatula as needed.
In the same small bowl/measuring cup, no need to clean it, combine the boiling water and espresso powder and add to the batter in the mixer bowl. Mix again on medium speed for 30 seconds, until smooth, scraping the bowl with a rubber spatula as needed. The batter will be quite thin
Pour the batter into the prepared pan and bake for 28 to 32 minutes, rotating at the halfway point. The cake is ready when a toothpick inserted in the middle comes out with a moist crumb or two. Let cool completely in the pan. Or let cool enough so you can comfortably handle it, and invert the cake on to a serving plate to continue cooling. If you would rather serve slices directly from the pan, let cool to room temp before frosting.
For the frosting: In a food processor fitted with the metal blade, or in a blender, process the berries until pureed. Place a sieve over a small bowl, pour the puree into the sieve, and using a rubber spatula, press the puree through the sieve to remove the seeds. Set aside.
Add the egg whites, sugar, corn syrup, cream of tartar, and salt in the now-cleaned bowl of the stand mixer and nestle it in a saucepan of simmering water over medium-high heat. Do not allow the bottom of the bowl to touch the water. Whisk until the mixture is just warm to the touch. Remove the bowl from the heat, add the puree and transfer it to the stand mixer fitted with the whisk attachment. Mix on high speed until stiff and glossy, about 7 minutes. Add the lemon juice and whisk a final time to combine.
Once the cake is cool, generously frost it and sprinkle the top with pink sanding sugar or sprinkles. The frosted cake is best served within a few hours with a few sliced strawberries on each plate - or, a scoop of strawberry ice cream - I mean this is a birthday cake, after all.