Preheat oven to 350°F (175°C) and grease a 9x13 inch baking dish.

Place potatoes in a pot, cover with salted water, and bring to a boil.

Reduce heat and simmer until potatoes are tender (about 20 minutes).

Drain potatoes and let them steam dry.

Heat vegetable oil in a skillet over medium heat.

Cook sausage in the hot oil for 5 minutes.

Add white and green onions and cook for another 5 minutes.

Stir in tomato, cumin, and thyme; season with salt and pepper.

Reduce heat to low and simmer for 10 minutes.

Place bread cubes in a bowl and pour milk over them.

Arrange half of the potato slices in the baking dish.

Spread sausage mixture over the potatoes.

Top with sliced eggs, half of the moistened bread, mozzarella cheese, remaining potato slices, remaining bread, and Parmesan cheese.

Bake in the preheated oven until heated through (about 30 minutes).