To keep the French toast warm between batches, preheat the oven to 200°F. Set a wire rack on top of a baking sheet, then place in the oven. Heat a large, sturdy skillet (a seasoned cast iron pan or large nonstick skillet both work) over low heat. Let warm up while you prepare the French toast custard.

In a wide, shallow bowl (a deep dish pie plate works well), whisk the whole eggs, egg yolk, milk, half-and-half, sugar, vanilla, cinnamon, and salt until the mixture is smoothly combined and a little foamy.

Add a thin slice of the butter to the warm skillet. Increase the heat to medium/medium-low so the butter melts and sizzles just a little bit.

Working 2 slices as most at a time, gently dip each piece of bread in the custard mixture on both sides, letting absorb for 3 or 4 seconds, then flip and let absorb on the other side for 3 or 4 seconds more. You want the bread to have a nice coating, but not to be soggy or soaked.

Lift a piece of bread out of the mixture, let the excess drip off, then place in the skillet. Dip and place more slices, until the skillet is full. Cook brioche French toast on each side until golden and the center springs back slowly when gently pressed, about 2 to 3 minutes per side depending upon your skillet's heat. Transfer to the oven to keep warm, or serve right away.

Add more butter to the skillet, then repeat with the remaining bread and custard. You'll likely need to continue adjusting the heat as you go so that the outside browns nicely without burning and the French toast cooks through. Enjoy immediately with your toppings of choice.