Place the apples, lemon zest, lemon juice, vanilla, and cinnamon in a large bowl. Stir to coat. Place in a large skillet over medium heat and cook, stirring occasionally, until the apples break down, 15 to 20 minutes. Stir in 4 tablespoons of the butter and ¾ cup of the sugar and continue cooking until the mixture has thickened, stirring occasionally, about 30 minutes.
Preheat oven to 375° F. Melt the remaining butter. Brush each bread slice generously on both sides with the butter. Press a third of the bread into an 11-by-7-inch baking dish. Sprinkle with a heaping tablespoon of sugar. Top with half the apple mixture. Repeat the process, pressing to compress, and ending with buttered bread. Whisk together the egg whites and the remaining sugar. Pour over the top of the dish and bake until golden and bubbling, about 25 minutes. Serve warm with whipped cream.