Mix together the chicken, cilantro, mayonnaise, salt, pepper to taste, the avocados and lime juice. Cover and refrigerate. For the dough: Mix together the water and salt in a large bowl, and then slowly add the cornmeal as you mix with your fingers. Mix and knead until it forms a soft dough. When you can pick up all of the dough in one mass, it's ready to form the arepas. Divide the dough into 8 balls, then gently flatten and form into discs.
Heat a lightly-oiled griddle or nonstick skillet over high heat. Preheat a grill to medium heat.
Cook the arepas on the griddle until they release from the cooking surface and are able to be moved, about 1 minute 30 seconds, and then flip and cook the other side for another 1 minute 30 seconds. Transfer the arepas to the grill and grill until lightly browned in spots, about 5 minutes per side. The arepas are ready when they sound like a drum when tapped.
Split the arepas in half, then scrape out and discard some of the middle. Stuff the arepas with the chicken mixture and eat while still piping hot.