Heat the oil in a large skillet over medium-high heat. Season the chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper and cook until browned and cooked through, 8 to 10 minutes per side. Transfer to 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, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Cook, tossing gently, until the spinach begins to wilt, 1 to 2 minutes. Serve with the chicken.