Adjust an oven rack to the middle position and heat the oven to 200 degrees Fahrenheit.
Arrange the bread on a wire rack set over a baking sheet and bake until slightly dried, about 15 minutes.
Melt 2 Tablespoons of the butter and whisk it together with the milk, egg, sugar, vanilla, cinnamon, and salt in a medium bowl.
Slowly whisk in the flour until smooth. Pour into a large, shallow dish.
Heat a 12 inch nonstick skillet over medium heat. Add 1 Tablespoon of the remaining butter and heat until just beginning to brown.
Lightly whisk the egg mixture and soak 2 pieces of the bread for about 15 seconds per side.
Remove the bread from the egg mixture, allowing excess to drip back into the dish, then add to the heated skillet.
Cook until golden brown on both sides, approx. 2 ½ minutes per side.
Repeat with remaining butter and bread.
Serve.
