In a medium skillet, heat the olive oil and brown the cubed chicken over medium-high heat until cooked through, about 5 minutes. Remove from the pan with a slotted spoon and cover with foil to keep warm.
Add the green pepper and pineapple chunks to the skillet, reserving the pineapple juice. Cook over medium heat until the peppers soften, about 2-3 minutes.
In a small mixing bowl, whisk together the pineapple juice, apple cider vinegar, honey, cornstarch and ginger paste.
Pour the mixture into the skillet and stir constantly until it begins to thicken. Add the chicken back to the skillet and cook until rewarmed then serve.