Combine shallot and lime juice in a medium bowl and mix well. Let sit for about 5 minutes (this makes the shallot taste milder). Add avocado, peppers, grapes and coriander and season with salt and pepper to taste. Cover and let marinate at room temperature for about 15 minutes.
Meanwhile, heat oil in a frying pan and brown the chicken over a high heat on all sides. Turn heat down to medium, add chicken seasoning and cook for a few more minutes until chicken is no longer pink in the middle. Remove from the heat and set aside.
Halve and toast the bagels, spread cream cheese on each half, top with the cooked chicken and sprinkle with cheese. Place back in the oven and grill until cheese has melted.