Preheat a grill to medium-high. In a medium saucepan, bring the juice, preserves and vinegar to a boil. Lower the heat and cook until thick and syrupy, 7 to 8 minutes.
Drizzle EVOO all over the chicken and season with salt and pepper. Grill until cooked through, 6 minutes on each side.
Coat the beets with EVOO and season with salt and pepper. Grill until well marked and tender, 10 to 12 minutes. Slice the beets and top with the cheese.
Brush the chicken with the apricot glaze and serve with the beets.