In a 12-inch cast-iron skillet, cook sausage over medium heat until browned and crumbly, 7 to 10 minutes. Using a slotted spoon, transfer sausage to a medium bowl, reserving drippings in skillet.
To skillet, add 2 tablespoons oil. Add rutabaga; cook, stirring frequently, for 2 minutes. Add sweet potato; cook until vegetables are just tender, stirring frequently, 6 to 10 minutes. Transfer rutabaga mixture to bowl with sausage.
To skillet, add remaining 1 tablespoon oil. Add bell pepper, onion, and garlic; cook until softened, 3 to 4 minutes, stirring frequently. Stir in vinegar, thyme, maple syrup, salt, and red pepper. Return rutabaga-sausage mixture to skillet.
Make 4 wells in rutabaga mixture; crack 1 egg into each well. Reduce heat to medium-low. Cover and cook until egg whites are opaque, 8 to 12 minutes. Garnish with red pepper and thyme, if desired. Serve immediately.
Recipe by Taste of the South at https://tasteofthesouthmagazine.com/rutabaga-breakfast-hash/