1. In large sauté pan over medium heat, add breakfast sausage, breaking into pieces, and cook until browned. Remove to paper towel–lined plate and reserve.
2. To same pan, add onions and cook for 10 minutes until soft and caramelized. Remove from pan and reserve.
3. Add vegetable oil to pan along with sweet potatoes. Cook over medium-high heat until sweet potatoes are browned and soft but not mushy.
4. Add Brussels sprout leaves, garlic and sage to pan, and cook for 3 minutes more. Add in reserved cooked sausage and caramelized onions and season with salt and pepper.
5. Fry eggs in separate sauté pan.
6. To serve: Plate hash in bowl and top with fried eggs.