Peel the sweet potato and cut into 1/2-inch pieces.
In a large heavy skillet, heat 2 tablespoons of the oil over medium high heat until hot, then stir in the onion, sweet potato, 1/2 teaspoon salt and 1/4 teaspoon pepper and cook, stirring occasionally, until golden and tender, about 8 minutes. Let the sweet potato mixture cool to warm.
Place each chicken breast, smooth-side-up, on a work surface. Place the palm of one hand on top of one breast, then insert a pairing knife to make a length-wise pocket in each breast. Divide the sweet potato filling between the breasts, stuffing them full. Reserve any leftover sweet potato to serve on the side. Rub the chicken with the remaining 1 tablespoon oil, then season all over with 3/4 teaspoon salt and 1/2 teaspoon pepper.
Preheat the oven to 425°F.
Place the chicken breasts on an oiled baking sheet and bake until cooked through, about 30 minutes. Let stand 5 minutes before slicing and serving.
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