Preheat oven to 425 F. Wrap all four heads of garlic in aluminum foil and place on the top rack of the oven to roast for 20 minutes, your garlic should be soft to the touch and easy to push out of the garlic skin into a sort of paste once finished roasting. Peel your potatoes and quarter them. Cover them with water, add in about 2 tbsp salt, and bring to a boil until the potatoes are soft and easy to pierce/mash. Your garlic and potatoes should be done around the same time.
In the meantime, grab a cutting board and your leeks. With a sharp knife, remove the dark green, tough part of the leeks and set aside (you can use this for soup stocks or bone broth). Slice the white and light green part of the leeks into thin round slices, making sure to discard the root end. Grab your saute pan and heat over medium high heat for about a minute. Add in the 2 tbsp butter and enough olive oil to cover the bottom of the pan generously. Add in your sliced leeks reduce the heat to medium. Scatter them evenly across the bottom and saute them for about 10 minutes, moving them every 2-3 minutes until they are soft and white stems are transluscent.
When your garlic is done, take three of the roasted heads and squeeze them out of the garlic skins into a small bowl. Mash with a fork until you have a smooth paste and pour into your saute pan with the leeks and saute together for about three minutes, until they are equally combined. Transfer the leeks and garlic to the dutch oven and add in the potatoes, fresh thyme, vegetable broth, and water. Bring to a simmer for five minutes, and then remove the sprigs of thyme.
Transfer the mixture to a blender and blend until it reaches the texture you want (you're the boss of whether you want it slightly chunky or smooth). Pour back into the dutch oven and bring back up to a simmer. Begin seasoning with salt and pepper, stirring for about 30 seconds in between tasting. Remember that potatoes absorb A LOT of salt, so it will take some generous salting to season your soup. A good indicator that you are done seasoning is that tasting makes you close your eyes. Once you've reached that point, move onto the ancho chili oil.