Grease a 9x13-inch pan. Layer brioche cubes evenly in the prepared pan.

Cook the bacon in a skillet over medium heat, stirring occasionally, until it is cooked through, about 10 minutes. Remove from heat, including any rendered bacon fat, and allow to cool slightly, about 5 minutes.

Heat a small amount of bacon fat in the same skillet over medium-high heat. Saute onion and bell pepper until soft, 5 to 7 minutes.

Beat eggs in a large bowl. Mix milk, Cheddar cheese, mustard powder, pepper, and salt into the beaten eggs. Mix in bacon and onion-bell pepper mixture when done. Carefully pour the egg mixture over the brioche bread cubes and give it a gentle stir so everything is evenly distributed. Gently press the cubes into the egg mixture as needed, so all the bread cubes are sitting in the liquid. Cover and refrigerate for 8 hours, to overnight.

Preheat the oven to 350 degrees F (175 degrees C).