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.