In large bowl, whisk together eggs, egg yolks, 2 tablespoons sugar, vanilla and salt. Stir in milk and cream. Add bread and press down, making sure all of the bread is covered by the custard. Allow bread to soak while cooking the apples.
Toss apples with lemon juice. In large skillet, melt butter over medium heat. Add apples, remaining 1/2 cup of sugar, brown sugar, cinnamon and cayenne pepper. Cook until apples are slightly tender, 8 to 10 minutes, stirring frequently. Cool 20 minutes.
Preheat oven to 325º F. Generously butter 9 x 13-inch casserole dish. Pour half of bread mixture into dish. Top with half the cooked apple mixture, then with 1 cup of cheese. Repeat layering with remaining ingredients. Cover casserole with waxed or parchment paper, then cover dish tightly with foil. Bake in water bath for 60 minutes. Remove foil and paper; return to oven. Increase oven temperature to 350º F. Continue to bake in water bath for 25 to 35 minutes more or until browned and puffed, and a butter knife inserted near center comes out clean. Remove from oven and allow to rest 20 minutes before serving. Sprinkle top with cinnamon and cayenne, if desired.