Preheat oven to 425º, ensuring that one rack is on the lowest level, and one rack is a bit closer to the broiler.
Add eggs, brown sugar, cinnamon, vanilla, and salt to a medium sized bowl, and whisk until thoroughly mixed. Make sure that the egg has fully incorporated into the blend of vanilla and dry ingredients.
Whisk the melted butter into the egg mixture, and add milk. Be sure to whisk all ingredients until well combined.
Coat a large, rimmed baking sheet in vegetable oil based cooking spray, and pour in custard mixture.
Place bread into the baking sheet, in a single layer, to allow it to begin soaking up the custard mixture. Allow it to soak for 30 seconds to 1 minute, and flip. Allow the other side to soak for about a minute as well, to absorb the remaining custard.
If the bread doesn't soak in all the custard, just pour the remaining liquid back into a measuring cup or the bowl you used earlier. (Make sure not to spill the bread!) Pour the extra custard back on top of each slice of bread, in small quantities, until you've used it up. Try not to pour it back onto the baking sheet, because we want the bread to be able to soak up everything!
Cook for 10-15 minutes on the lower rack, until the bottom of the bread is golden brown.
Move the tray of French toast onto the rack closest to the broiler, and turn the broiler on. Allow the bread to sit under the broiler until it's golden brown on top, up to 4 minutes. Be sure to monitor it the whole time, to avoid burning it! Don't walk away during these final, crucial moments.
Remove from the oven, and flip the slices. Allow to sit for just a minute or so, and then serve!