Preheat the oven to 425°F (220°C). Line a baking sheet with parchment paper.
Scrub the sweet potatoes and pat them dry. Pierce each potato several times with a fork, then rub lightly with olive oil, salt, and pepper.
Place the sweet potatoes on the baking sheet and bake for 40–45 minutes, or until they are very tender when pierced with a knife.
Meanwhile, heat a large skillet over medium-high heat. Add the ground turkey and cook for 5–7 minutes, breaking it apart with a wooden spoon as it browns.
Add the onion, bell pepper, carrot, and mushrooms. Cook for 5–6 minutes, until the vegetables soften and the turkey is fully cooked.
Stir in the garlic and tomato paste and cook for about 1 minute.
Add the tomato sauce, ketchup, Worcestershire sauce, vinegar, brown sugar, smoked paprika, chili powder, salt, pepper, and water. Stir well to combine.
Reduce the heat to medium-low and simmer for 8–10 minutes, stirring occasionally, until the sauce thickens and coats the meat.
Remove the sweet potatoes from the oven and let them cool for 2–3 minutes. Cut a deep slit lengthwise down the center of each potato and gently open it.
Fluff the inside of each sweet potato with a fork, then spoon about ¾ cup of the Sloppy Joe mixture over each potato.
Top with green onions, pickles, pickled red onions, and shredded cheddar if desired. Serve warm.