Preheat the oven to 350 degrees . Coat a 9-by-13-inch baking pan with cooking spray; line with parchment paper. In a bowl, combine the boiling water and espresso. Whisk in 1/4 cup cold water, then the oil, whole egg and vanilla.
In a large bowl, whisk together 1/2 cup granulated sugar, the flour, brown sugar, cocoa powder, baking powder, baking soda and salt. Whisk the egg mixture into the flour mixture until just combined. Spread the batter evenly in the prepared pan. Bake until a toothpick inserted into the center comes out clean, 18 to 20 minutes. Let cool completely on a rack, then invert onto a baking sheet. Discard the parchment paper. Line the pan with another piece of parchment paper.
Using a 2 1/2-inch cutter, cut the cake into 8 rounds and transfer to the prepared baking pan; freeze for 30 minutes. Using a 2-inch-wide ice cream scoop, top each cake with a scoop of sorbet; cover the pan with plastic wrap and freeze for at least 3 hours or overnight.
In a small saucepan, combine the remaining 1 cup granulated sugar with 1/4 cup water and cook, stirring, over medium heat until the sugar is dissolved. Boil until slightly reduced, about 3 minutes.
Using an electric mixer, beat the egg whites at medium speed until soft peaks form, 1 to 2 minutes. With the mixer on low, drizzle in the sugar syrup, then beat on high speed until the egg whites are stiff and glossy, about 8 minutes. Pipe or spread the meringue over each cake to cover. Freeze for up to 1 hour.
Preheat the broiler. Broil the meringue-covered cakes until lightly browned, about 2 minutes.