Season chicken with salt and pepper. Melt 1 Tbsp. butter with 1 Tbsp. oil in a large skillet over medium-high heat. Add two chicken breast halves; cook until no longer pink in center, about 3 minutes per side. Remove to a plate; cover with foil. Repeat with remaining butter, oil and chicken.
Pour out all but 1 Tbsp. fat from skillet. Sauté onion for 3 minutes. Add vinegar, raise heat to high and boil until reduced, about 2 minutes. Reduce heat to medium-low; add spinach and mandarin oranges. Sauté until spinach has wilted, about 1 minute. Season with salt and pepper.
Slice chicken. Divide spinach mixture among 4 plates. Serve with chicken and top with peanuts.