You can prepare the sausage breakfast casserole up to a day ahead and let the bread soak overnight in the refrigerator. (If it is refrigerated overnight, it may need to cook a few minutes longer.)
In a large skillet over medium-high heat, cook sausage, stirring occasionally, and breaking up meat with a spoon, until browned, 8 to 10 minutes. Add onion; cook. Stirring, until soft, 8 to 10 minutes. Stir in sun-dried tomatoes. Transfer to a plate; let cool until barely warm.
Butter an 8-inch-square baking dish. In a large bowl, whisk together eggs, milk, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Arrange half the bread in a slightly overlapping layer in bottom of dish. Scatter half the sausage mixture and then half the cheese over the bread. Repeat with remaining bread, sausage, and cheese, pressing firmly. Pour egg mixture over layers. Cover and refrigerate at least 20 minutes or up to overnight.
Preheat oven to 350 degrees. Bake, uncovered, until puffed and golden and a toothpick inserted into center comes out clean, 35 to 45 minutes.
Alternate the ingredients as you would with a lasagna, then saturate them with the egg-and-milk mixture.