Place the mushrooms face down on baking sheets. Brush with a little oil and season with salt and pepper. Flip the mushrooms over and remove the stalks. Brush with oil, drizzle with balsamic vinegar and season with salt and pepper.
Bake the mushrooms for 6 minutes at 200C/390F (180C/355F fan). Pour off any excess liquid. (It's delicious mushroom stock, so save it for use in another recipe.) Put the mushrooms back in a low oven to keep warm while you prep the mince.
Warm a teaspoon of oil in a pan over a medium heat. Add the minced garlic and diced red onion. Fry until soft and just starting to brown. Add the mince and fry until browned. Drain off any excess liquid. Stir in the torn coriander.
Spoon the mince mix into the warm mushrooms. Sprinkle with crumbled feta, season and serve two mushrooms per person.