Heat the oil in a fry pan over medium heat. Add the onions and sauté until translucent.
In a large sauce pan, add the water, potatoes and a tsp. of salt. Bring to a boil, then remove the foam that forms on the top.
Stir in the cooked onions and milk. Bring to a boil, then add the cheese, salt, pepper and bijol.
Reduce heat to low and simmer until potatoes are soft and falling apart, about 40 minutes.
Mash the potatoes a bit with the back of a spoon to thicken the soup. Leave the soup a little chunky.
Add salt and pepper as needed. Serve the soup with a few slices of avocado in each soup bowl.