Preheat grill or grill pan. Spray with cooking spray.
In a nonstick frying pan sprayed with cooking spray, cook mushrooms and onions over medium-high heat for 4 minutes, or until softened and browned. Remove from heat. Stir in feta.
In a bowl, stir together beef, chives, oregano, barbecue sauce, breadcrumbs, garlic, and egg. Stir in onion mixture. Form into 5 patties.
Grill patties over medium-high heat for 3 to 5 minutes per side, or until cooked through. Patties can also be placed on a baking sheet sprayed with cooking spray and baked in centre of preheated 450°F/230°C oven for 10 to 15 minutes, or until cooked through, turning once.