Place the sweet potatoes in a large pot. Add enough cold water to cover and bring to a boil.
Add 1 teaspoon salt, reduce heat, and simmer until tender, 14 to 16 minutes. Reserve ¼ cup of the cooking water, drain the sweet potatoes, and return them to the pot. Mash with the reserved cooking water.
Meanwhile, heat 1 tablespoon of the oil in a large skillet over medium heat. Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper.
Cook the chicken until golden brown and cooked through, 7 to 8 minutes per side. Transfer to plates.
Wipe out the skillet. Heat the remaining 3 tablespoons of oil over medium-high heat. Add the shallots, rosemary, ½ teaspoon salt, and ¼ teaspoon pepper and cook, stirring, until the shallots are tender, 3 to 4 minutes.
Serve the chicken with the potatoes and drizzle with the shallot mixture.