Pour the hot water in a large bowl and then add the achiote OR turmeric, salt, and sugar. Crush achiote paste with a fork and blend well.
Slowly add the corn meal and stir it with a wooden spoon or spatula. When the corn meal has absorbed the water, knead it with your hands until the dough is smooth and without any lumps.
Roll the dough between pieces of parchment and use a cookie cutter to make 4-inch (10 cm) rounds that are about ¼-inch (6 mm) thick.
Gently place the arepa in oil that has been pre-heated at 325F (160 C) and let it sink to the bottom.
Once the arepa rises to the surface use a large spoon to repeatedly coat the top surface of the dough with more oil for about six to eight minutes, until the arepa puffs up. Remove the puffed arepa from the oil and drain it on paper towels. Repeat the process until all are fried.
Once the arepa is cool enough to handle, cut it in half to make a bun.
Heat oil in non-stick skillet over medium heat. Fry the eggs to preferred temperature (such as sunny side up), season with salt and pepper.
Portion the cooked eggs in the arepa buns, garnish with avocado and pickled red onion and serve immediately.