Preheat the oven to 400 degrees F (200 degrees C).

Heat olive oil in a large skillet over medium heat. Add sweet potato; cover and cook, stirring occasionally, until tender, 8 to 10 minutes. Transfer to a large bowl.

Cook and stir sausage in the same skillet over medium-high heat until crumbled and browned, 5 to 7 minutes. Add to sweet potato in the bowl.

Cook and stir onion and red bell pepper in the same skillet until tender, about 3 minutes. Season with salt and pepper. Add mushrooms and kale; cook until kale softens, about 3 minutes more. Transfer to the bowl.

Whisk eggs, water, thyme, salt, and pepper together in a small bowl. Stir into sausage mixture. Pour into a large baking dish.

Bake in the preheated oven until sweet potato starts to brown, 20 to 25 minutes. Let stand for 5 minutes. Garnish with green onion.