Fill a large, straight-sided skillet or Dutch oven with 2 inches of water; bring to a boil. Add white vinegar.

Meanwhile, heat oil in a large nonstick skillet over medium-high heat. Add shallot and garlic and cook, stirring, until fragrant, about 1 minute. Stir in zucchini and tomatoes and cook, stirring occasionally, until the zucchini is tender, about 10 minutes. Remove from the heat; stir in 1 tablespoon basil, balsamic vinegar, salt and pepper.

Meanwhile, reduce the boiling water to a gentle simmer; the water should be steaming and small bubbles should come up from the bottom of the pan. Crack each egg into a small bowl and slip them one at a time into the simmering water, taking care not to break the yolks. Cook for 4 minutes for soft set, 5 minutes for medium set and 8 minutes for hard set. Using a slotted spoon, transfer the eggs to a clean kitchen towel to drain.

To serve, top each muffin half with some of the vegetable mixture, an egg, a sprinkling of cheese and the remaining basil.