Place the bread slices on a baking sheet and bake at least 30 minutes at 300-degrees to dry out the bread. Flip halfway through the baking. Remove from the oven and cool. Set aside.
In a medium bowl, stir together the cream cheese and blueberry emulsion until smooth and creamy.
Spread the flavored cream cheese liberally on one side of each slice of bread.
Top three of the cream cheese slices of bread with fresh blueberries, pressing them gently into the cream cheese.
Top each of the three sandwich bottoms (with the blueberries) with a slice of turkey and a slice of ham.
Add shredded cheddar cheese evenly to each sandwich on top of the lunch meat and then press the sandwich top (cream cheese layer down) on top of the filling.
Place the assembled sandwiches into the 300-degree oven for approximately 10-15 minutes, or until the cheddar cheese has melted.
Remove the sandwiches from the oven and press them firmly to adhere all the ingredients together.
In a shallow dish or bowl, whisk together the eggs, milk, and salt.
Place each sandwich in the egg mixture and allow the bread to sit for 30 seconds. Carefully flip the sandwich and allow it to sit in the egg mixture on the other side for at least 30 seconds.
Remove the Monte Cristo sandwiches to a parchment sheet and allow to sit for 10 minutes.
Heat a large skillet over medium/high heat. Add butter to the skillet and gently fry each sandwich until golden on both sides, flipping carefully.
Serve the sandwiches warm with a light dusting of confectioners sugar if desired.