Combine the stock, olive oil, vinegar, garlic, thyme, and salt in a large plastic bag. Arrange the mushroom caps in one layer in the marinade, turning once to coat. Seal the bag, pressing out excess air. Marinate the mushrooms at room temperature, turning occasionally, for about 1 hour.
Prepare broiler or grill. Away from the heat source, lightly coat the grill rack or broiler pan with cooking spray. Position the cooking rack 4 inches (10 cm) from the heat source.
Arrange the mushrooms on the rack or broiler pan and grill or broil, turning often, until tender, about 7 minutes.