In the bowl of a stand-mixer fitted with the paddle attachment, beat together the cream cheese, powdered sugar, vanilla, and milk until smooth and combined. Scrape the sides of the bowl as needed.
In a bowl, whisk together the eggs, milk, vanilla, and sugar until completely combined. Place a skillet over medium heat. Melt 2 teaspoons of butter in the pan. Submerge a slice of bread into the custard mixture to thoroughly coat. Place the bread into the pan and cook until golden and crisp on each side, about 2-3 minutes per side, being sure to cook through. Repeat with remaining slices of bread, adding more butter to the pan as needed between each slice.
Place one slice of French toast on a serving plate. Spread half of the cream cheese mixture onto the French toast. Layer with half of the strawberries. Press another slice of French toast on top, pressing down slightly, sandwiching the filling in between the slices. Repeat with remaining ingredients. Dust each plate with powdered sugar and serve promptly while hot with maple syrup.