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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.
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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.
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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.
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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.