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. Add the tomatoes, season with ¼ teaspoon each salt and pepper, and cook until just tender, 1 to 2 minutes per side. Transfer to a plate.
Add the remaining tablespoon oil to the skillet and heat over medium-high heat. Add the mushrooms, thyme, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, tossing occasionally, until golden brown and tender, 6 to 7 minutes.
Meanwhile, crack 4 of the eggs into separate custard cups. One at a time, slide each 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.
Dividing evenly, top the bread with the tomatoes, mushrooms, eggs, Parmesan, and ¼ teaspoon each salt and pepper. Sprinkle with the chives.