Brush mushrooms and tomatoes with oil and sprinkle with salt and pepper. Arrange mushrooms, stemmed sides down, and tomato halves, cut sides up, on a broiler pan.
Broil vegetables about 6 inches from heat, turning over mushrooms halfway through cooking, until tender and tomatoes are slightly charred, about 7 minutes total. (Leave broiler on.)
While vegetables are broiling, fill a deep 10-inch skillet with 1 1/4 inches cold water.
Add white vinegar and bring to a simmer.
Break 1 egg into a cup, then slide egg into simmering water. Repeat with remaining eggs, spacing them in skillet, and poach at a bare simmer until whites are firm but yolks are still runny, 2 to 3 minutes.
Gently transfer eggs with a slotted spoon to paper towels to drain and season with salt and pepper.
Sprinkle stemmed sides of mushrooms with balsamic vinegar, then put 1 tomato half, cut side up, on each mushroom and top with an egg. Cover eggs with cheese slices and broil until cheese is just melted, about 1 minute.
Sprinkle with chives and serve immediately.
· The eggs in this recipe are not fully cooked, which may be of concern if salmonella is a problem in your area. You can substitute pasteurized eggs (in the shell) or cook eggs until yolks are set.