Heat the achiote oil in a large nonstick skillet over medium heat. Add the chopped green onions and cook until soft and fragrant, about 3 to 5 minutes.

Stir in the drained 30-ounce (1-pound) can of hominy. Cook, stirring frequently, for 3 to 5 minutes so the hominy absorbs the flavor and warms through. Season with salt to taste.

In a small bowl, lightly beat the eggs with a pinch of salt. Reduce the heat to medium-low, pour the eggs over the hominy, and gently stir and fold, as if making scrambled eggs mixed into the hominy, until the eggs are just set but still soft and moist, about 3 to 4 minutes.

Turn off the heat. Stir in the chopped cilantro. Sprinkle the shredded cheese over the top, cover the pan, and let it sit for 2 to 3 minutes so the cheese melts.

Serve warm, on its own or with hot sauce/ají on the side.