To make the marinade, microwave lime in a small bowl to release essential oils, and set aside until cool enough to handle. Through the feed opening of a running blender add, 1 at a time, half of the papaya (reserving the other half), chile pepper, vinegar, salt and pepper. Then with the blender still running, squeeze in the juice from the microwaved lime.
Place abalone in a nonreactive container, pour marinade over, cover and refrigerate for 6 to 8 hours.
Use marinade to make a sauce by adding it to a pan over medium-high heat. Bring to a boil and let reduce and thicken. In a small bowl, create a slurry by gradually adding 1 tablespoon water to cornstarch. Whisk slurry into sauce, and add reserved papaya slices and papaya seeds. Adjust seasoning with salt and pepper, remove from heat, cover and keep warm.
Grill abalone over medium-high heat until opaque. Remove from heat and let rest.
Serve with sauce and sprinkle with chopped cilantro.