Add your cashews and carrot to a small pot and cover with water. Bring the pot to a boil on the stovetop, and allow the mixture to boil for 15 minutes (or until the carrots and cashews are both very soft and easily pierced with a fork).
Drain the water and add the cashews and carrot to a high powered blender. Add all other ingredients to the blender as well. Blend until very smooth.
Pour your blended mixture into a small pot or sauce pan and stir constantly as you bring it to medium-high heat on the stovetop. When you first pour in your mixture it will be very runny, but after a couple of minutes it will begin to thicken. Once your mixture is thick and cheesy, remove from the heat. (If you'd like your cheese sauce to be a bit thinner, feel free to add a couple splashes of water and incorporate as needed before removing from heat). This entire stovetop process should take about 3-5 minutes.
Spread on whatever you'd like and enjoy! (If using for a vegan grilled cheese, please see the notes below on how I recommend assembling your sandwich!*)