Combine butter, brown sugar, and corn syrup in a small saucepan over medium heat; stir until melted and smooth.
Pour the caramel mixture into a 9x13-inch baking dish, spreading evenly across the bottom.
Arrange brioche slices in the dish in a single layer over the caramel.
In a medium bowl, whisk together eggs, half-and-half, vanilla, salt, and nutmeg until well combined.
Pour the egg mixture evenly over the bread slices.
Cover the dish with plastic wrap and refrigerate for at least 8 hours or overnight.
Preheat oven to 350°F (175°C) and remove the dish from the refrigerator to take the chill off.
Bake uncovered for 35 to 40 minutes until the bread is puffed and golden brown.