Preheat the oven to 350°F.
In a large skillet over medium heat, heat 1½ tablespoons of olive oil.
Sauté the onion and garlic until the onion is slightly translucent, about 5 minutes.
Add the sweet potato, and cook until fork-tender, about another 15 minutes. Spoon the mixture into an 8-by-12-inch baking dish.
Add the remaining 1½ tablespoons of olive oil to the skillet, and sauté the mushrooms until browned, 5 to 7 minutes.
Season with salt and pepper, and add them to the baking dish as a second layer.
Add the sausage to the skillet, and cook until no more pink remains, 7 to 10 minutes.
Season with salt and pepper, and add to the baking dish as a third layer.
In a large bowl, whisk the eggs well, season with salt and pepper, and pour over the casserole mixture in the baking dish.
Bake for about 1 hour, or until the eggs are no longer runny.
Serve hot, garnished with a sprinkle of sliced scallion.