Prepare Mushroom Meatballs: Heat oil in a large nonstick skillet over medium-high. Add mushrooms and onion; cook 5 minutes, until soft. Add garlic, thyme, and oregano; cook 1 minute. Transfer to a large bowl and let cool 10 minutes. Add soy sauce, egg, oats, breadcrumbs, milk, salt, and pepper; stir to combine. Refrigerate mixture for 30 minutes.
Prepare Spanish Olive-Tomato Sauce: Heat oil in the same skillet you used for the mushrooms over medium heat. Add onions; cook 5 minutes, until softened. Add garlic, tomato paste, and red pepper flakes; cook 2 minutes, stirring occasionally. Add tomatoes, broth, paprika, sugar, salt, and pepper; bring mixture to a boil. Reduce to a simmer and cook, uncovered, 15 minutes, using a wooden spoon or potato masher to lightly crush tomatoes, until sauce thickens. Add meatballs to sauce, and sprinkle olives and feta cheese overtop. Serve over pasta or orzo, couscous, or with crusty bread.