In large nonstick skillet, heat oil and butter over high heat until sizzling. Add mushrooms; stir-fry until tender, about 5 minutes. Add garlic and thyme; stir-fry for 1 minute. Turn heat to low to keep warm.
Meanwhile, bring large saucepan of water to simmer; add vinegar. Crack eggs, 1 at a time, into small bowl; tip into simmering water. For runny eggs, poach for about 3 minutes. Remove eggs with slotted spoon and gently pat dry.
Meanwhile, toast bread; place slice on each plate. Season mushrooms with salt and pepper; spoon onto toast. Top each with poached egg. Sprinkle with pepper and thyme leaves.