Heat 1 tablespoon oil in a large skillet over medium heat. Add sliced shallots, 2 teaspoons thyme, and garlic; cook 3 minutes or until shallots are tender. Add wine, 1/4 teaspoon salt, 1/4 teaspoon pepper, and tomatoes; cook 5 minutes or until liquid is reduced by half.
Weigh or lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, milk, remaining 1 tablespoon oil, remaining 1/4 teaspoon salt, remaining 1/4 teaspoon pepper, cheese, and egg yolk in a large bowl, stirring with a whisk until smooth. Add 2 teaspoons vinegar and baking powder, stirring well to combine.
Beat egg white in a large bowl with a whisk until stiff peaks form (do not overbeat). Fold egg white into batter. Preheat waffle iron; coat with cooking spray. Spoon about 1/3 cup batter per 4-inch waffle onto waffle iron. Cook 5 minutes.
Fill a skillet two-thirds full with water; add remaining 1 teaspoon vinegar; bring to a boil. Reduce heat to a simmer. Break each whole egg into a custard cup, and pour each gently into pan; cook 3 minutes or until desired degree of doneness. Carefully remove eggs using a slotted spoon. Place 1 waffle on each of 4 plates; top evenly with tomato mixture and eggs. Sprinkle with remaining 1 teaspoon thyme.