Fill a saucepan with enough water to cover the eggs by 1 inch. Add 1 tablespoon of salt and bring to a gentle boil. Once the water is boiling, gently lower the eggs, being careful not to break them. Reduce heat to a gentle boil and cook the eggs for 14 minutes.
With a slotted spoon, carefully remove the eggs from the boiling water and submerge them immediately into a large bowl of ice water. Let cool completely, about 15 minutes, then peel under cool running water.
In a medium bowl, stir together the Greek yogurt, Dijon mustard, vinegar, curry, yellow mustard, cayenne, and 1/4 teaspoon salt.
Cut the eggs in half from tip to tip. With a small spoon, gently scoop out the yolks and add them to the bowl with the curry mixture (for a very smooth filling, press the egg yolks through a sieve and into the bowl). Set the egg whites aside.
Mash the yolks into the filling with the back of a fork until smooth. Taste and season with additional salt and cayenne as desired.
Fill the hollow egg whites with the curry mixture (I place the mixture in a ziptop bag, then snip off the corner to pipe). Refrigerate until ready to serve. Just before serving, sprinkle with paprika and chives.
Notes
MAKE AHEAD: Make the hard boil the eggs ahead and keep them unpeeled in the refrigerator for up to 1 week before mixing the filling and assembling as directed.
TO STORE: Assembled deviled eggs will keep for up to two days in the fridge.