In a small bowl whisk together the honey and the lime juice. Combine the pineapple, cucumber and mint in a medium bowl, pour the dressing over and toss to combine. Set aside.
Heat 2 teaspoons of olive oil in a large skillet over a medium-high heat. Add the chicken breasts and cook for about 4 minutes on each side, or until browned and cooked through. Transfer the chicken to a plate and cover with foil to keep warm.
Add the remaining 2 teaspoons of oil to the pan. Stir in the scallions, pepper, garlic, ginger, allspice, and thyme. Cook for 30 seconds over a medium heat. Add the chicken broth and soy sauce and cook until liquid is reduced by half, about 3 minutes. Stir in the lime juice. Put the chicken back in the pan and coat well with the sauce.