Melt butter in a large skillet over medium heat. Once melted, add the brown sugar and a pinch of cinnamon. Stir with a wooden spoon until the sugar dissolves and forms a sauce. Add the banana slices and toss to coat.
Cook until the bananas soften slightly and blend into the sauce. Transfer to a bowl and allow to cool completely. To speed this up, place the bowl in the refrigerator. Once cooled, add the cream cheese and mash gently until combined.
Spoon about 1/6 of the cream cheese mixture onto one slice of bread and top with another slice to form a sandwich. Repeat with the remaining bread slices. Refrigerate the prepared sandwiches while you prepare the egg batter to help the filling stay firm.
For the egg batter, whisk together the eggs, milk, ground cinnamon, and vanilla extract if using in a medium shallow bowl. Set aside.
In a large skillet, melt 1 to 2 tablespoons of butter over medium heat. While the butter melts, dip a few sandwiches (as many as will fit in the skillet) into the egg mixture, soaking each side for about 3 seconds. Place them in the skillet and cook each side for 2 to 3 minutes, or until golden brown. Wipe the skillet clean between batches if needed.
Keep the cooked sandwiches warm on a baking sheet in an oven set to 200 degrees F until ready to serve. Serve warm with butter, maple syrup, fruit, whipped cream, or powdered sugar.