Pre-heat the 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, 7-8 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, 10-12 minutes. Top with the cheese.
Brush the chicken with the apricot glaze and serve with the beets.