Preheat oven to 350 degrees F and lightly butter your baking dish.
Add the babka bread cubes to your 10-inch cast-iron skillet.
In a medium-sized mixing bowl, whisk together the milk, granulated sugar, light brown sugar, unsalted butter, kosher salt, and vanilla extract until well combined. Pour most of the custard, about ¾ of the entire thing, evenly over the bread cubes.
Let the bread soak in the custard for 30 minutes to 1 hour. Pour on the reserved custard to moisten any dry pieces of bread on top.
Bake in the preheated oven for about 45 minutes. The bread pudding will be done when a butter knife or cake test inserted in it comes out clean.
Brown Butter Bourbon Sauce
Melt butter in a stainless steel or light-colored medium-sized saucepan. Stir the butter around constantly, and in about 3 to 5 minutes the butter will begin to smell toasted and nutty and the milk solids will have turned a golden brown.
Whisk in flour and whisk constantly for one minute to cook out raw flour taste and ensure nothing sticks to the bottom on the pan.
Splash in a small amount of milk at a time, about ¼ cup, whisking after each addition until the milk is absorbed into the flour. It will be a thick, clumpy goop at first, and that's how you know you're doing it right. Keep adding the milk, whisking constantly, until all the milk is added and the consistency is smooth.
Mix in the brown sugar and kosher salt. Let cook, stirring occasionally so the bottom doesn't burn, until the sauce is thickened, about 15 minutes.
Remove from heat and add your bourbon and vanilla extract. Pour over the bread pudding while warm.
Keep the bread pudding in an air-tight container or wrapped well with plastic wrap in the refrigerator for up to 5 days.