In a medium pot, bring 4 cups (960 ml) of water to a boil. Add the cashews and carrot and boil for 10 minutes, or until the carrot is fork tender.
Use a slotted spoon to transfer the cashews and carrot to a high-speed blender. Do not discard the cooking water.
To the blender, add the nutritional yeast, tapioca starch, lemon juice, salt, garlic powder, turmeric, and 1 cup ( 240 ml) plus 2 tablespoons of reserved cooking water (discard any remaining water). Blend the sauce for 2-3 minutes on high speed, until completely smooth.
Pour the “cheese” sauce back into the pot and stir with a spatula over low heat. Clumps should begin to form. Continue stirring for 2-4 minutes, or until the sauce is thickened and stretchy. Remove the pot from the heat and set aside.
Butter each slice of bread on 1 side. In a medium nonstick skillet, heat a dollop of vegan butter over medium heat. Once the butter begins to bubble, add a slice of bread, butter side down, and use a spatula to spread cheese sauce on top. Top with another slice of bread, butter side up. Cook for 2-4 minutes, or until the bottom of the bread is golden brown. Flip and cook for 2-4 minutes more, until the other slice of bread is golden brown as well, then remove from pan. Repeat with the remaining ingredients.
Slice the sandwiches in half and serve immediately.