To prepare the caramelized Apples: Combine the sugar and 1/4 cup of water in a large heavy sauté pan over medium heat. Stir until the sugar dissolves and the syrup comes to a simmer. Allow the syrup to boil over medium-high heat without stirring, brushing down the sides of the pan and swirling the pan occasionally to ensure it cooks evenly, for about 6 minutes, or until it begins to turn golden brown. Remove the pan from the heat and add the apples and butter and swirl until the butter melts. Cook over medium heat for 2 minutes, or until the apples are just tender. Meanwhile, to make the French toast: Using a fork, beat the eggs in a 13x9 baking dish to blend. Place the slices of brioche in the eggs and let stand for 5 minutes, or until the eggs are absorbed, turning the brioche slices once. Stir the sugar and cinnamon on a large plate; set aside. Melt the butter on a heavy large griddle pan over medium heat. Add the brioche slices to the hot pan and cook for about 2 minutes per side, or until golden brown on the outside and heated through. Immediately place the hot French toast in the cinnamon-sugar and turn to coat completely. Divide the French toast among 4 serving plates. Spoon the apples over the French toast, and drizzle the caramel sauce from the pan over the apples and around the French toast. Top with a dollop of crème fraîche and serve immediately