Prick each sweet potato all over with a fork and microwave until fork tender, rotating about halfway through, about 10 minutes. Set aside and keep warm.
Preheat the oven to 400 degrees F. Meanwhile, heat the olive oil in a medium skillet over medium-high heat. Add the sausage and cook, stirring occasionally, until it just starts to brown and crisp up, 5 to 6 minutes; set aside.
Cut the potatoes in half lengthwise. Scoop the flesh into a large bowl, leaving a 1/4-inch border all the way around. Place the potato skins on a baking sheet and set aside. Add the cooked sausage, 1/2 teaspoon salt and a few grinds of pepper to the bowl with the potato flesh and stir until well combined. Divide the potato mixture evenly among the 4 potato halves and make a well big enough to hold an egg in the middle of each. Crack an egg into each and bake until the egg whites are set and the yolks are still runny, 15 to 17 minutes. Serve with the salsa verde.
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