Fill a large, deep skillet with 3 inches of water. Add the vinegar and bring to a bare simmer.
Meanwhile, heat 1 teaspoon of the oil in a large nonstick skillet over medium-high heat. Season the tomatoes with 1/4 teaspoon each salt and pepper and cook until just tender, 1 to 2 minutes per side; transfer to a plate.
Add the remaining tablespoon of oil to the skillet and heat over medium-high heat. Add the mushrooms, thyme, 1/2 teaspoon salt, and 1/4 teaspoon pepper and cook, tossing occasionally, until golden brown and tender, 6 to 7 minutes.
Meanwhile, poach the eggs in 2 batches: Crack each of 4 eggs into separate custard cups, then, one at a time, slide gently into the water in the deep skillet. Cook for 2 to 3 minutes for slightly runny yolks and remove with a slotted spoon. Repeat with the remaining 4 eggs.
Top the bread with the tomatoes, mushrooms, eggs, and Parmesan. Season with 1/4 teaspoon each salt and pepper and sprinkle with the chives.