Fill a medium saucepan with water and a splash of vinegar, bringing it to a very gentle simmer over medium heat.
In a small microwave-safe bowl, combine the nonfat Greek yogurt and the shredded cheddar cheese.
Heat the yogurt and cheese mixture in the microwave for 30 to 45 seconds, stirring halfway through, until the cheese is completely melted and the sauce is smooth.
Whisk the sea salt and black pepper into the warm yogurt-cheese sauce until well incorporated.
Toast the slice of white bread until it is golden brown and crisp, then place it in the center of a plate.
Crack each egg into a small ramekin or cup to ensure the yolks remain intact.
Gently slide the eggs one by one into the simmering water and poach for 3 to 4 minutes until the whites are fully set but the yolks are still runny.
Carefully remove the poached eggs from the water using a slotted spoon and place them on a paper towel to drain any excess moisture.
Arrange the poached eggs on top of the toasted white bread.
Pour the warm, velvety yogurt-cheese sauce generously over the eggs and serve immediately while hot.