Combine the quinoa is a saucepan with 2 cups water with a good pinch of salt. Bring to a boil, cover,and simmer 10-15 minutes until the quinoa is fluffy and tender.
In the meantime, heat 1 teaspoon of the oil in a large skillet over medium heat. Add the chard and red pepper flakes and cook until wilted and tender, about 8 minutes. Season with salt and pepper and remove to a plate.
Add the remaining oil into the skillet and add the garlic over medium heat. When it is slightly golden, add the mushrooms. When they release their moisture, about 3 minutes,season with a pinch of salt and pepper. When they are tender, about 5 minutes,stir in the reserved chard and the thyme and heat through. Stir in the quinoa and divide among 4 bowls or plates. Top with shaved Parmesan.