Place potato and carrots into a medium saucepan and cover with water. Bring to a boil. Lower heat and allow to cook at a low boil until tender, about 15 to 20 minutes. Remove from heat and drain into a colander. Transfer to a food processor and blend until completely smooth, stopping to scrape down sides of bowl as needed.

Coat the bottom of a large pot with oil and place over medium heat. Add onion and sauté until softened, about 5 minutes. Add garlic and cook 1 minute more. Add mushrooms, in as even a layer as possible, and allow to cook until slightly browned, about 5 minutes.

Add broth and rice to pot. Stir to incorporate. Raise heat and bring to a simmer. Cover and allow to simmer until rice is fully cooked, about 45 minutes. There should be some liquid remaining in the pot when rice is finished cooking.

Uncover pot and stir in potato mixture, broccoli, nutritional yeast, lemon juice and miso, if using. Thin with about 2 cups of milk, using a bit less if mixture seems too thin, more if it seems too thick. Stir and bring to a low simmer. Cook just until broccoli is tender and bright green, about 3 minutes. Season with salt and pepper. Serve.