In a blender or mini-food processor combine onion, garlic, oregano, marjoram, smoked paprika, rosemary, salt, pepper, olive oil, and vinegar. Cover and process until combined (mixture will be puréed but not totally smooth).
Add the marinade to the mushrooms and gently stir to coat. (Take care not to break up the portobello mushrooms.) Cover and chill at least 6 hours or up to 24 hours.
Transfer mushrooms to a colander to drain off any free liquid. Preheat a 10-inch grill pan or cast iron skillet over medium high heat. Lightly brush pan with oil. Add mushrooms to pan in batches and cook until nicely browned on both sides, about 3 minutes per side for trumpet mushrooms (2 minutes per side for portobello mushrooms).
Serve mushrooms wrapped in warm, Greek style pitas with shredded lettuce, sliced tomatoes, sliced red onion, sliced cucumber, kalamata olives, feta, and tzatziki sauce. Finish off each one with a sprinkle of smoked paprika.
