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Heat oven to 375 degrees. Butter an 11-by-11-inch baking dish. Fill dish with brioche cubes and set aside.
In a large bowl, beat eggs well. Whisk in half-and-half, salt, pepper, nutmeg, cayenne and scallions. Pour mixture over cubes in baking dish.
Sprinkle ham and cheese on top of mixture, and press down to make sure everything is submerged. Set baking dish on a rimmed baking sheet.
Transfer baking dish and baking sheet to the middle shelf of the oven, and bake for 45 minutes, or until a skewer inserted in the middle of the dish emerges dry. Let rest for 10 minutes or so before serving.