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Transfer chicken to a plate and drain off all but 1 tablespoon of the drippings. Return to stove on medium heat, add onions, and season with salt and freshly ground black pepper. Cook until golden and softened, about 3 minutes.
Add potatoes and broth and scrape the pan to incorporate any browned bits. Add chicken, bring to a boil, and then reduce heat to medium-low. Cover and cook until chicken is cooked through, about 30 minutes. Remove chicken to a plate and cover.
Bring sauce to a boil over medium-high heat and cook until slightly reduced, about 10 minutes. Taste and adjust seasoning, as desired. Add chicken, turn to moisten in sauce, and serve.