Fill a medium saucepan with water and bring to a rolling boil over high heat.
Carefully lower the eggs into the water using a slotted spoon and reduce the heat to a simmer for exactly 6 and a half minutes.
Prepare an ice bath in a small bowl while the eggs cook.
Remove the eggs and immediately place them in the ice bath for 3 minutes to stop the cooking process.
In a small bowl, whisk together the Greek yogurt, lemon juice, half of the chives, sea salt, and black pepper until smooth.
Toast the sourdough bread slices until golden brown and crisp.
Spread the savory yogurt mixture generously over each slice of toast.
Gently peel the soft-boiled eggs, slice them in half to reveal the jammy yolks, and place them on the toast.
Garnish with the remaining fresh chives and an extra pinch of black pepper if desired.