In a skillet or Dutch oven with a cover, heat 1 tablespoon of olive oil over medium heat. (If you plan to bake the casserole the same day, preheat oven to 350 degrees F.)
Add the sausage and cook until it is no longer pink, about 10 minutes.
Remove from the skillet and set aside.
Add the remaining olive oil.
Add the chopped onions and saute until they are slightly tender, 3 minutes.
Add the sweet potato, stir to coat with the oil, cover the skillet and cook for 5 minutes. Test with a toothpick - the sweet potatoes should be slightly softened but not mushy. If not, continue to cook for 2-5 minutes.
Add the broccoli florets and cook for 2-3 minutes more, uncovered.
Return the sausage to the skillet and mix to combine.
Season with salt and black pepper to taste and remove from the heat.
Pour the sausage mixture into a lightly greased 9 x 13-inch baking dish (unless you plan to use the Dutch oven for the entire dish, in which case just let the Dutch oven cool a bit).
In a bowl whisk the eggs.
Add the parsley and seasonings to taste.
Mix to combine.
Pour the eggs evenly over the sausage mixture.
Chill, covered, overnight. The next day, remove from the refrigerator for about 15 minutes before baking to take the chill off the casserole.
Bake in preheated oven, uncovered, at 350°F for 35 minutes (40 minutes if refrigerated overnight). The casserole is done when the eggs are set in the middle.
Sprinkle with extra fresh parsley, slice and serve.