Combine the yogurt, garlic, dill, 2 tablespoons olive oil, 1/2 teaspoon salt and a few grinds of pepper in a medium bowl; stir until smooth. Spread the yogurt on 4 plates.
Bring about 3 inches of water to a boil in a wide pot. Add the vinegar and reduce the heat to medium to maintain a gentle simmer. Crack 4 eggs into individual small bowls or ramekins. Carefully lower each egg into the water. Cook until the whites are set but the yolks are still runny, about 3 minutes. Remove the poached eggs with a slotted spoon and blot the bottoms dry on a kitchen towel. Divide among the plates. Let the water return to a simmer and repeat with the remaining 4 eggs; add to the plates. Season the eggs with a pinch each of salt and red pepper flakes.
Toast the walnuts in a small skillet over medium heat, stirring, until golden, about 4 minutes. Stir in a pinch of red pepper flakes, then stir in the butter until melted. Spoon over the eggs.
Toss the watercress with the remaining 1 tablespoon olive oil and a pinch of salt in a medium bowl; divide among the plates. Serve with the bread.