Arrange the mushrooms in a shallow dish. Whisk together the balsamic vinegar, oil, cumin, salt, pepper, paprika and garlic in a bowl, then pour over the mushrooms. Leave to marinate for 10 minutes, turning the mushrooms halfway through to ensure they are evenly coated in the marinade.
Meanwhile, prepare the chimichurri. Add the parsley, garlic, onion, chilli flakes, oil, salt, pepper and lemon zest and juice to a bowl and whisk well. Add the avocado and toss to combine. Set aside.
Heat a barbecue grill plate or a large chargrill pan to medium–hot.
Transfer the mushrooms to the barbecue or pan and cook for 4–6 minutes, turning halfway through and brushing over any remaining marinade as you go, until they are caramelised and a deep golden brown.
To serve, divide the mushrooms among plates and top with the avocado chimichurri.