1. To make Cashew Ricotta: Preheat oven to 400°F. Slice off top of garlic head, drizzle with 1 tsp. oil, and wrap in parchment paper, then in foil. Roast 1 hour, or until soft.
2. Rinse and drain soaked cashews, and transfer to food processor. Add remaining 1 Tbs. oil, nutritional yeast, lemon juice, salt, and vinegar. Squeeze roasted garlic from cloves into food processor, and blend until smooth. Transfer to medium bowl.
3. To make Linguine: Bring beets and enough water to cover by 1 inch to a boil in large saucepan. Cover, reduce heat to medium-low, and simmer 35 to 40 minutes, or until beets are tender. Drain beets, and slip off skins under cool running water. Coarsely chop beets, and set aside.
4. Heat 1 Tbs. oil in large skillet over medium heat. Add onion, and sauté 8 to 10 minutes. Reduce heat to medium-low, and cook 15 minutes more. Stir in balsamic vinegar and mirin. Cook 2 minutes, or until liquid evaporates. Transfer onion mixture to food processor, add beets, and purée until smooth. Season with salt and pepper, if desired, and transfer to medium saucepan. Keep warm.
5. Cook pasta according to package directions. Drain, reserving 3 Tbs. pasta water. Return pasta to pot; toss with beet sauce, remaining 1 1/2 tsp. oil, and reserved pasta water. Top each serving with 2 Tbs. Cashew Ricotta, and sprinkle with chopped parsley.