Preheat the oven to 400°F.

Line a baking sheet with foil. Prick the sweet potatoes 3 to 4 times each with the tines of a fork and place on the baking sheet. Brush the potatoes all over with 2 tablespoons of the oil and sprinkle all over with 1 teaspoon of the salt.

Bake the potatoes until the skin is crisp and browned and the tip of a paring knife slips easily into the flesh, 45 minutes to 1 hour.

Meanwhile, make the filling. In a large skillet, heat the remaining 2 tablespoons oil over medium heat. Add the garlic, jalapeño, and onion, and sauté until softened, about 3 minutes. Sprinkle the flour over top, then stir together and cook until lightly browned and bubbling, about 1 minute. Slowly add the chicken broth while whisking constantly. Cook, stirring, until slightly thickened, about 1 minute. Stir in the chicken, green chiles, pepper, paprika, and remaining 1 teaspoon salt. Remove from the heat and stir in the sour cream and lime juice.

When the potatoes are cooked, let them cool for 5 minutes to make them easier to handle. Switch the oven to broil.

Split open each potato lengthwise and lightly mash the flesh with a fork. Divide the chicken filling between the potatoes and sprinkle the top of each with the cheese. Broil the potatoes until the cheese is melted and starting to brown, 2 to 3 minutes. Top with cilantro and serve with lime wedges for squeezing over the top.