Heat the oil in a large skillet over medium-high heat. Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper.
Cook chicken until browned and cooked through, 8 to 10 minutes per side. Transfer to individual plates.
Spoon off and discard all but 1 tablespoon of the fat and return the skillet to medium heat. Add the shallots and cook, stirring, until they begin to soften, 2 to 3 minutes.
Stir in the wine and sour cream. Add the spinach and ¼ teaspoon each salt and pepper.
Cook, tossing gently, until the spinach begins to wilt, 1 to 2 minutes. Serve with the chicken.