Put the potatoes in a large pot and cover with cold water, making sure the potatoes are covered with at least 1" of water.
Meanwhile, in a small saucepan over low heat, add the almond milk, coconut milk, garlic, olive oil, coconut oil and salt (I recommend using 2 teaspoons and adding more later, if desired). Once it's all melted, turn the heat off (but keep the saucepan on the stove so it stays warm).
Drain the potatoes and return the pot (with the potatoes in it) back to the stove. Let any remaining moisture cook off for 1-3 minutes, shaking the pot occasionally. This helps with the texture of the mashed potatoes.
Mash the potatoes with a potato masher, food mill or ricer and gradually add the milk mixture, bit by bit (while also tasting to see if you want more salt), until well combined with the potatoes. The more you mess with the potatoes, the gummier they'll be. I used a potato masher, then added the liquid, and once I couldn't get any further with the masher (the mashed potatoes looked very, very soupy) and I couldn't get out any more lumps, I switched to a hand mixer and used that just a bit until the lumps were totally gone and the potatoes were fluffy and not at all soupy.
Cover cooled mashed potatoes and refrigerate for up to 4 days.