Cook the beans: Rinse the beans. Place the beans in a pot and fill the pot with enough water to cover the beans by one inch. Bring to a boil, cover, turn off the heat, and allow beans to soak for one hour.
Drain and rinse beans. Place beans back in the pot and cover with enough water to cover by one inch. Bring to a boil. Skim foam from the surface. Add 1/4 onion and garlic. Reduce heat and simmer until tender, about 2 hours. Stir in 1 1/2 teaspoons salt. Remove and discard the onion and garlic.
Make the refried beans: Heat a large skillet over medium-low heat. Add one tablespoon of avocado oil and heat.
Add onions, 1/2 teaspoon of salt, and 1/4 teaspoon of pepper. Cook and stir until the onions are just starting to brown.
Add two tablespoons of oil to the pan and increase heat to medium-high. Scoop 4 cups of beans out of the cooking liquid and place drained beans in the skillet. Cook until beans begin to split open and break apart, stirring occasionally.
Add 1/2 cup of cooking liquid and 1/2 teaspoon salt. Mash with a potato masher or place in a blender for a smoother consistency. Return to the pan if blended.
Add more cooking liquid until desired consistency. Stir in one tablespoon of butter. Taste and adjust salt if needed.
