In a small bowl, stir together cream cheese and powdered sugar until well combined. Divide cream cheese mixture among 4 slices of bread. Spread in an even layer, leaving a ¼ to ½ inch border around the edges. Top with remaining 4 slices. Set aside.
In a medium sized bowl, add eggs, milk, vanilla, ½ teaspoon cinnamon, nutmeg, and salt. Whisk together until well combined.
In a shallow bowl or on a plate, stir together the sugars and cinnamon. Set aside.
Heat a large griddle or pan over medium heat. Melt butter in bottom of pan to coat. Working in batches, take bread/cream cheese sandwiches that you prepared and dip into egg mixture. Let sit for a few seconds to soak, then flip and soak the second side for a few more seconds. Remove from egg mixture, and place on griddle. Let cook until golden brown on one side, then flip. Continue to cook until golden brown on the other side, bread isn't soggy, and egg coating is cooked. (This may take a little longer than regular french toast since it is thick/stuffed).
Once done cooking, immediately remove french toast and place on cinnamon sugar mixture, flipping to coat both sides. Continue with remaining french toast. Serve warm.