Boil the eggs: In the bottom of a medium saucepan, carefully arrange the eggs in one single layer and cover with at least 1-inch of cold water. Set on high heat to bring to a rolling boil. As soon as the water is boiling fast and hard, turn the heat off and cover the saucepan. Leave the eggs in the covered pan for exactly 10 minutes.
Run the cooked eggs under cold water: Drain the cooked eggs and run them under cold water until they have fully cooled (this will stop the eggs from cooking and will make them easier to peel).
Peel and halve the eggs: Peel the eggs and slice them in half from top to bottom.
Prepare the filling: Using a small spoon, scoop out the cooked egg yolks and put them into the bowl of a small food processor fitted with a blade. Add the yogurt, mustard, garlic powder, and a big pinch of kosher salt. Close the lid of the food processor and pulse several times until the mixture is fluffy and well-combined.
Fill the egg halves with the deviled egg mixture: Transfer the egg mixture to a piping bag fitted with a tip and squeeze to fill each egg cavity with the egg mixture (or you can use a spoon).
Serve: Sprinkle with the paprika, then garnish with chives and capers (if using). Serve immediately, or store in an airtight container in the fridge for up to 2 days.
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Notes
Believe it or not, you shouldn’t reach for your freshest eggs when you make this recipe–older eggs are easier to peel when they’re hard boiled.
If you don’t have a food processor, you can mash the filling in a mixing bowl until well combined and smooth.