Place the six eggs in a saucepan and cover them with at least 1 inch of cold water. Bring the water to a full boil, then remove the pan from the heat, cover it, and let the eggs stand for 10–12 minutes.

Transfer the cooked eggs immediately to an ice bath and chill for at least 10 minutes. The cold water stops carryover cooking and helps the shells separate more cleanly.

Peel the eggs, rinse away any shell fragments, and cut each egg lengthwise in half. Carefully lift the yolks into a bowl, keeping the whites intact for filling.

Mash the yolks until smooth, then stir in the Greek yogurt or sour cream, Dijon mustard, pickle juice or lemon juice, kosher salt, black pepper, and optional chopped pickles or relish. Taste and adjust the mustard, acid, and salt while keeping the mixture thick enough to hold its shape.

Spoon or pipe the yolk mixture into the egg-white halves. Dust the tops with paprika and add chopped chives or fresh dill if desired.

Refrigerate the filled deviled eggs until serving. If possible, chill them for 20–30 minutes so the filling firms and the flavors come together.