Whisk eggs and milk together until pale yellow and frothy. Combine bread, white Cheddar, marjoram, salt, and pepper together in a large bowl. Stir egg mixture into bread mixture until evenly coated. Grease a 1-quart baking pan; fill evenly with bread mixture. Refrigerate until set, 2 to 4 hours.
Preheat oven to 325 degrees F (165 degrees C).
Cover filled baking pan with aluminum foil; place on a rimmed baking sheet. Pour enough boiling water into baking sheet to reach halfway up the sides of baking pan.
Bake in preheated oven for 35 minutes. Remove aluminum foil and bake until golden brown, about 15 minutes more. Remove from water bath carefully; cool until set, about 5 minutes.