Preheat oven to 350F and butter an 8x10 casserole or baking dish. Whisk together eggs, sweetened condensed milk, milk, and cinnamon.
Gently dip a slice of brioche bread in the egg mixture. I find that brioche absorbs the egg mixture very fast compared to sourdough, so you don't want it to sit in the mixture; just a quick dip to make sure it's covered, and then transfer it to a greased 8x10 baking dish.* (Important: See note.)
In a small saucepan, add sauce ingredients: butter, brown sugar, vanilla, and melt over medium heat until creamy and just a little bubbly, about 6-8 minutes. Don't cook for too long, or it will turn into a tacky caramel instead of a sauce.
Pour warm sauce over the top of dipped and arranged brioche.
Bake uncovered** for 30 min with the 8x10 baking dish on top of a cookie sheet! You may experience some bubbles that run off the side, and you don't want this delicious breakfast to make a mess in your oven.
Let cool for 10-15 minutes. Then, top with powdered sugar and mixed berries.
* As you're staggering the slices of bread, you may find it’s easiest to start at the back of the dish and move towards you. Don’t fret too much if you have to readjust the slices. Even if they fall apart, this will all turn into one delicious custard-like loaf once it goes in the oven! (You'll see from the step photos I readjusted my bread slices about halfway through.)** If you notice an edge starting to burn or get too dark for your liking, snip off a little piece of tinfoil to cover it while it finishes baking so it doesn’t burn.