Toasting give the delicate bread a firmer texture, so the milk and eggs don't make it mushy. You can also use toasted French bread cubes, but the slight sweetness of the Hawaiian bread brings out the salty ham and nutty cheese flavors. Use the remaining rolls to make mini ham and cheese sandwiches.
Heat a small nonstick skillet coated with cooking spray over medium-high heat. Add onions, ham, and garlic; sauté 5 minutes. Remove from heat; cool.
Preheat oven to 350°.
Arrange bread cubes on a baking sheet. Bake at 350° for 15 minutes or until lightly browned, turning occasionally.
Combine milk and next 5 ingredients (milk through nutmeg) in a large bowl, stirring with a whisk until well blended. Stir in ham mixture. Add bread, tossing gently to coat.
Arrange half of bread mixture in an 8-inch square baking dish coated with cooking spray. Sprinkle with half of cheese; top with remaining bread mixture. Bake at 350° for 25 minutes. Sprinkle with remaining cheese; bake an additional 20 minutes or until set.