Method

1. Fill a large high-sided saucepan with enough water to reach a depth of 6 inches. Stir in the vinegar and bring the water to a near simmer over medium heat.

2. Meanwhile, place a large heavy nonstick sauté pan over medium heat until hot. Add the butter and 2 tablespoons of the oil, then add the shallots and garlic and sauté until fragrant, about 30 seconds.

3. Add the mushrooms and season with salt. Sauté until the mushrooms are tender, about 5 minutes. Remove the pan from the heat and stir in the parsley. Season to taste with salt and pepper.

4. When the vinegar-water is at a near simmer, crack 1 egg into a coffee cup or a small bowl and then gently transfer the egg to the barely simmering water. Repeat with 3 more eggs.

5. Cook the eggs until the whites are set but the yolks are still runny, about 3 minutes. Using a slotted spoon, carefully remove the eggs from the simmering water and set them on a paper towel to drain the excess water.

6. Set the warm toasted bagel halves cut side up on 4 plates. Set the eggs on the bagels and season with salt and pepper. Top with the sautéed mushrooms. Drizzle with the remaining oil and serve immediately.