SOFTEN: Heat a saucepan or dutch oven over low-medium heat. Dice the onion, then smash and peel the whole garlic cloves. Add the oil, diced onion and smashed garlic to the pan, stirring through the oil to coat. Allow the onion and garlic to soften, stirring occasionally to avoid any burning or coloring. Meanwhile, chop the cauliflower into evenly sized florets and add to the pan along with the salt and broth. Increase the heat to medium-high and bring the broth to a simmer. Cook until the cauliflower is fork tender, about 20 minutes.
BLEND: Remove the pan from the heat (turn off the burner) and carefully transfer it to a trivet and use an immersion blender to blend everything together until you have a smooth, creamy soup base. Add the oregano leaves and return the pan to the heat.
SIMMER: Turn the heat to medium and bring the soup back up to a simmer. If the soup is thicker than you prefer, add a little extra broth until the soup is the texture that you like. Add the pulled pork and cook until the pork is hot all the way through before serving.