Make the masa for the churritos: Combine tortilla flour with grated cheese, egg, salt and pepper to taste, and the achiote dissolved in 3/4 cup water. Knead the doughly lightly until soft and pliable but not sticky. Cover with a cloth and let rest.
To make the soup: heat 3 tablespoons of the oil in a saucepan over medium heat. Add the onion and bell pepper and cook until the vegetables are tender.
Add the milk, 3 cups of water, and salt and pepper to taste. Add the bouillon cube, if using. Take 1/2 cup of the masa and dissolve it in 1/2 cup water. Add to the soup.
Simmer until lightly thickened, and taste adding more salt or pepper.
In a deep frying pan heat the remaining oil over medium heat to about 365 degrees. Knead the reserved masa (add a little water if needed) and form 2-inch logs, about 3/4-inch thick, or make 2-inch mini tortillas. Fry them in the hot oil until golden. If you want to add some of them to the soup, you can do so.
Transfer the fritters to a plate covered with paper towel.
Just before serving, drop the eggs in a thin stream in the hot soup, stirring all the time.
Put some masa fritters in each soup bowl, cover with soup and garnish with hierbabuena or parsley. Serve immediately.