In a large bowl, combine the chicken and the lime juice. Let stand while you gather the rest of the ingredients. Then add the onion, coriander stems, half the coriander leaves, the peppers, garlic, broth or water with tomato paste, and the potatoes. Toss with salt and pepper.
Generously film with oil the bottom of a 12-inch skillet with an ovenproof handle. Heat over medium-high. Turn the contents of the bowl into the skillet and spread it out. Cook, adjusting heat as needed, to slowly crisp the ingredients (8 to 10 minutes). Use a spatula to turn pieces if they threaten to stick.
When vegetables are starting to brown, spread out the mix over the skillet so the bottom of the mix can crisp. Once golden brown, use the back of a tablespoon to make 4 small wells in the top of the hash. These will hold the eggs.
Slip the hash under the broiler for about 2 minutes to start it browning. Carefully remove it to a heatproof surface, break an egg into each of the wells, sprinkle with salt and pepper and broil another 1 minute, or until the egg white is firm and yolk is creamy.
Serve the hash hot, sprinkled with the rest of the fresh coriander leaves.