Whisk the eggs, milk, vanilla, cinnamon, and a pinch of salt until they run in an uninterrupted line through the tines of a fork.
Warm a non-stick pan over medium-low heat. Add 3 tablespoons of butter and a 1/2 tablespoon of cooking oil. Heat until the butter foams.
Dip the slices of bread in the batter, covering both sides, allowing them to soak briefly. The slice should feel heavy, like a wet sponge, but shouldn't drip like a soaking wet sponge.
Place the slices in the pan and sprinkle the tops with a pinch of sugar. Cook until the slice has browned on the bottom. Flip and sprinkle the browned tops with a second pinch of sugar.
When you press your finger in the slice's center and it springs back, turn it over again and cook briefly to caramelize the last sprinkle of sugar. Repeat for all slices, adding more oil and butter as needed. Slice and serve warm.
*If you want to substitute the eggs, add the milk and spices to a small pot. Whisk in a tablespoon of cornstarch until it's completely dissolved, then place the milk over medium heat. Continue whisking until the milk comes to a boil. Continue to cook and whisk for 1 minute at a full boil. Take the thickened milk off the heat and cool before dipping the slices of bread into it.