Preheat oven to 375 degrees F (190 degrees C). Grease a baking sheet.
Remove stems and scrape out gills of mushrooms. Brush both sides of mushrooms with olive oil.
In a medium skillet, heat 1 tablespoon olive oil over medium-high heat. Add red onion and garlic; cook and stir until softened, about 3 minutes. Stir in oregano, thyme, and salt; cook for 1 minute more. Add quinoa; cook for 2 minutes to heat through.
Divide quinoa mixture among mushroom caps. Sprinkle feta cheese over each cap.
Bake in preheated oven until mushrooms are tender and cheese is melted, about 20 minutes.