For Arepas: Adjust your oven racks to the upper- middle and lower-middle positions and preheat the oven to 350°F. Whisk together the corn flour, sugar, and salt in a large bowl. Set aside.
Whisk the butter into 21/2 cups of water, then add it to the flour mixture, stirring until well combined. The dough will start out loose but the flour quickly absorbs the liquid. Start to knead the dough in the bowl and once it becomes very soft and doesn’t stick to your hands, after about 8 minutes, the dough is ready to be shaped (if, while kneading, the dough seems too stiff and breaks apart, add a few tablespoons of hot water; if it is too sticky, add a little more corn flour).
Divide the dough into 16 equal balls and flatten each between your palms into a 3 1/2- to 4- inch patty that’s about 1/3 inch thick (for a less rustic- looking arepa, press the arepa into a disk using a flat- bottomed plate; you can wet your hands with a little water if the dough is slightly sticky).
Heat 1 tablespoon of the vegetable oil in a large nonstick skillet over medium- low heat for 2 minutes. Add 3 or 4 arepas to the pan (depending on how big your pan is); the arepas should sizzle as they hit the skillet. Cook the arepas until they’re golden and have a nice crust, 6 to 8 minutes. Flip them and brown the other side for an additional 6 to 8 minutes. Transfer the arepas to a rimmed baking sheet and set aside. Repeat with the remaining 1 tablespoon oil (if the pan is dry) and the remaining arepa dough disks (you’ll probably need to use two baking sheets to bake the arepas).
Bake the arepas until they puff up, 20 to 30 minutes, switching the pans so the top baking sheet is on the bottom and the bottom moves up to the top, midway through cooking. Remove from the oven and set aside for 5 minutes
For Reina Pepiada: Place the chicken in a large bowl. Add the onions, mayonnaise, lemon juice, and cilantro and salt and pepper to taste and stir to combine. Gently stir in the avocado, taste, and adjust the seasonings as needed.
For Plating: Slice a slit into the top edge of each arepa and gently wiggle a paring knife into the arepa, creating a pocket. Divide the chicken salad among the arepas and serve.