Make the custard for the Pain Perdu but whisk together the milk, heavy cream, eggs, sugar, Armagnac, and vanilla extract until the mixture is completely homogenous.
Place the bread in in a deep dish or tray that is just large enough to hold the bread in a single layer and cover with the custard. If your dish is too large the custard won't soak into the bread completely. If you don't have a suitable dish, you can use a sealable plastic bag and press out the excess air. Cover and refrigerate for a day, turning the bread over a few times in between.
Preheat the oven to 450 degrees F (230 C). Remove the soaked Pain Perdu from the refrigerator and flip one more time.
Mix 1 tablespoon of sugar with 1 tablespoon of flour and sprinkle half the mixture onto the tops of the bread using a small sieve(such as a tea strainer) to ensure the flour gets sprinkled evenly.
Add the butter to a cast-iron skillet and heat over medium heat.
When the butter has melted and the foaming subsides, add the bread with flour-sprinkled side down.
Dust the Pain Perdu with the remaining flour/sugar mixture and fry until it's well browned on one side (about 5 minutes). If your heat is up too high it will burn, so if it looks like it's browning too quickly, turn the heat down.
Flip the bread over and put the pan in the oven.
Bake for 8-10 minutes. Keep a close eye on it as the sugar will burn easily. You want the surface of your Pain Perdu to be very dark, but not burnt.