In a medium saucepan, bring several cups of water to boil.

Place prepared potato, carrot, and onion into the boiling water, and cook until potatoes are fork tender, or about 10 minutes.

While your potatoes, carrot, and onion are cooking, place 2/3 cup water (preferably water from your pot of cooking/cooked vegetables), cashews, coconut milk, yeast flakes, lemon juice, salt, and garlic powder into your blender. Blend on high until everything is creamy smooth.

When potatoes are fork-tender, place potatoes, carrot, and onion into your blender--using a slotted spoon to avoid adding excess water to your cheese.

Blend everything on high until smooth.

You can use this cheese as is (pretty warm) to make your grilled cheese sandwiches, or you can place it in the fridge for a few hours or days to cool and thicken up slightly--which makes a less messy grilled cheese sandwich. If you're like me, you don't mind ooey gooey all-over-the-place cheese, and the hot stuff works just fine.

Slather your cheese sauce on 2 thick slices of bread, slap 'em together, and brush the outsides of your sandwich with either coconut oil (or other heat-safe oil of your choice) or vegan butter. (I use Earth Balance, generally.)

Place your sandwich in a preheated pan or griddle on medium heat. Usually, my sandwiches only need about 30 seconds to 1 minute on each side to be nice and crispy. Check your first sandwiches frequently, as they can burn quickly without warning.