Preheat the oven to 400F.
Pour the tomato sauce and diced tomatoes into a saucepan, season with salt and pepper, and bring to a boil over medium-high heat. Stir in the quinoa, reduce the heat, cover, and simmer gently, stirring occasionally, until the liquid is absorbed and quinoa is tender, about 15 minutes. Let stand, covered, for 5 minutes. Fluff with a fork, taste and adjust the seasoning with salt and pepper, and stir in the basil. Cover to keep warm until ready to serve.
Divide the sausage into 8 portions. Cut the cheese into pieces about the size of a quarter, 1/4-inch thick. Carefully form one portion of sausage around a piece of cheese into a patty, being sure to encase the cheese completely and sealing the patty. Repeat with the remaining sausage and cheese.
Heat the olive oil in a large non-stick skillet over medium heat. Add the sausage patties and cook until golden, 4-5 minutes. Flip the patties and continue cooking until browned, 3-4 minutes more. Transfer the patties to a baking sheet, cover loosely with foil, and bake in the oven until the sausage is cooked through, 6-8 minutes. Keep warm until ready to serve.
Meanwhile, pour off all but 2 tablespoons of the grease from the skillet and return the pan to medium-high heat. Add the onions and cook, stirring frequently, until softened and beginning to brown, about 5 minutes. Add the mushrooms and more oil, if needed; season with salt and pepper, and cook until the mushrooms are soft and beginning to brown, about 5 minutes more. Transfer the onion/mushroom mixture to a bowl. Add 1 tablespoon oil to the pan and return it to the heat; add the greens, season with salt and pepper, and cook, tossing the greens with tongs, until wilted, about 5 minutes. Add the chicken stock or water if the pan is dry. Return the mushroom mixture to the pan with the greens and toss to combine.
To serve, spoon about 1 cup of the stewed quinoa into a thin layer covering each of 4 dinner plates. Divide the greens and mushrooms between the plates, and top each plate with 2 cheesy sausage patties.