In a large bowl, whisk together the eggs, creamer, and vanilla until fully combined. Pour the mixture onto a large plate with edges, such as a pie plate or baking dish. Dip the bread slices into the mixture, being sure to coat both sides evenly. In a small bowl, whisk together sugar and cinnamon; set aside.
Melt 2 tablespoons of the butter in a large skillet over medium heat, and pan-fry two slices of battered bread at a time until golden brown, about 2 minutes per side. When the second side is golden brown, generously sprinkle cinnamon and sugar onto each piece and flip again. Let the sugar caramelize for 30 second to 1 minute; repeat process on the other side. Remove from the skillet and transfer to a plate; repeat process with remaining butter, battered bread and cinnamon sugar, two slices at a time.