Add the sweet potato, cauliflower florets, and carrots to a large pot. Cover with water and place over medium heat. Cover the pot with the lid and bring the water to a boil and cook until the vegetables are easily pierced with a fork.
Drain the vegetables well and transfer to a high speed blender along with remaining ingredients. Blend until the nacho cheese is smooth and creamy.
Add additional oil, brine, salt, or hot sauce to taste. Serve while warm and creamy. Enjoy!
Notes
Do not use a ninja blender with the blade up the stem. This blender is absolutely awful for making creamy vegan sauces because it is incapable of blending anything completely smooth. Because the blades go up the stem, it is only capable of very finely chopping things and any sauce you make in this blender will always have a slight grit. Blenders with the blade at the bottom, on the other hand, create a vortex formation that ensures everything is blended into a true liquid.
Do not use a food processor. For the same reason I do not recommend the ninja.
Invest in a Vitamix blender. It may counterintuitive to recommend you a blender that is a few hundred dollars, but frugal cooking is not about being as cheap as possible. It's about knowing when it makes sense to invest money and cook SMART. I purchased a Vitamix blender in 2016 for $199 from Costco and it still works perfectly. Easily one of the best investments I've ever made in my kitchen.
Worst case scenario, use a single serve blender cup. You may need to blend in a couple of batches and then combine everything in a larger bowl, but this is the best option if you do not have a high speed blender (this is what my recipe photographer did for this recipe and it worked great!).
If you are making a double batch, blend a little at a time. All of the ingredients fit in a 64 ounce Vitamix blender canister, but it is quite heavy on the motor. I like to add the ingredients a little at a time. This makes it way easier to blend without getting stuck.