Combine vegetable broth and plant milk in a medium saucepan. Bring to a gentle boil over medium heat. Add cornmeal and stir well with a fork or whisk. Reduce to low heat.
Simmer for 5 minutes, stirring frequently to prevent buildup and scorching on the bottom of the pot. It can be helpful to use a stiff spatula for this.
Stir in the nutritional yeast, cheese, and salt.
If the consistency is too thick for your liking, stir in more vegetable broth, plant milk, or water, 1/4 cup at a time, until it is thin enough. Each time you add liquid, stir for 30 seconds over low heat, so that the consistency is even and the grits are heated through.
If the consistency is too thin for your liking, continue to simmer over low heat until the desired consistency is reached.
Taste the grits and add salt to taste. You can add more cheese if you like, as well.
Serve hot, topping with 1/4 tablespoon of plant-based butter (if using) on each serving.