Heat oven to 425° F.
On a rimmed baking sheet, toss the potatoes, onion, 2 tablespoons oil, ½ teaspoon salt, and ¼ teaspoon pepper. Roast until tender, 20 to 25 minutes.
Meanwhile, heat the remaining tablespoon oil in a skillet over medium-high heat.
Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper and cook until browned and cooked through, 5 to 6 minutes per side.
In a large bowl, toss the warm potato mixture with the spinach, lime juice, and ¼ teaspoon salt. Serve with the chicken.