Preparation

• Position a rack in the center of the oven and heat the oven to 200°F. Have ready a large rimmed baking sheet.
• Set a strainer over a bowl. Put the zucchini in the strainer (in batches, if you like), and squeeze it well with your hands to extract some of the liquid. Reserve the liquid.
• Set aside 2 Tbs. of the scallions. In a large bowl, combine the flour, baking powder, 1 tsp. salt, and 1/8 tsp. pepper. Add the zucchini, cheese, and remaining scallions, and toss to coat in the flour. Beat 1 egg with enough of the zucchini liquid to yield 2/3 cup; add water if necessary to make 2/3 cup. Add to the flour mixture, and stir to make a thick batter.
• Heat a large nonstick skillet over medium-low heat. Add enough oil to coat the bottom of the pan (about 2 Tbs.). Scoop 1/3-cup portions of the batter onto the skillet and spread with the back of a spoon to form pancakes about 1/2 inch thick. Cook until golden brown on one side, about 5 minutes. Flip and cook until the center is cooked through, about 4 minutes more. Transfer the cooked zucchini cakes to a baking sheet and keep warm in the oven. Repeat with the remaining batter, adding oil to the pan between batches.
• Fill a 3-quart saucepan with 2 inches of water and bring to a boil. Add the vinegar and reduce to a simmer. One at a time, crack the remaining 4 eggs into a small bowl and gently slide each egg into the water. Poach until the whites are set, 2 to 3 minutes. With a slotted spoon, remove the eggs from the water in the order they were added and gently blot dry with a paper towel.
• Serve the zucchini cakes with the poached eggs, and sprinkle with the reserved scallion.