Special equipment: 11 x 17 x 1-inch baking tray, the bottom lined with parchment paper
Preheat the oven to 350 degrees F.
In a bowl of an electric mixer, combine the egg yolks, 1 cup of the sugar, salt, and vanilla. Using the whisk attachment on high speed whip until thick, 3 to 5 minutes. Reduce to medium speed and slowly add the hot water. Scrape down the sides of the bowl. Increase to high speed and again whip until thick. Fold in the flour and then the melted butter. Set aside.
In a clean bowl of an electric mixer using the whisk attachment, whip the egg whites on medium speed until frothy. Increase to high speed and slowly add the remaining 1/2 cup sugar. Whip the egg whites until stiff and then fold them into the reserved batter. Gently spread the batter in the prepared baking tray. Bake until the cake is golden brown and springs back lightly when touched, about 20 to 25 minutes. Cool the cake in the pan. Remove the cake from the pan by running a knife along the inside edge of the pan and inverting it onto a work surface. Carefully peel off the parchment paper.
The genoise can be made a day or two in advance. Store wrapped in plastic wrap at room temperature. It also freezes well.