Place the saffron in a skillet and bring to medium-high heat. Gently toast until the skillet is too hot for you to touch the center with your finger, about 1–2 minutes.
Transfer the saffron to a mortar and pestle, and grind it into a fine powder.
Pour 3 tbsp. of slightly warm water into the mortar. Set aside.
To make aioli, place the egg yolk in a medium bowl and add a pinch of salt. Whisk the yolk and start to drizzle the olive oil in very slowly. The yolk and the olive oil will start emulsifying. Continue to whisk vigorously. When it starts to get thick, pour a very small amount of the saffron liquid into the aioli to soften it and continue whisking. Keep adding olive oil until it thickens again. Add more saffron water to your liking.
Remove the cooked yolks from the whites. Set aside egg white halves. In a medium bowl, smash the yolks with a fork until smooth. Add the capers, cornichons, half the chives, and shallots. Mix well.
Add 2 to 4 tablespoons saffron aioli and mix well. For more saffron flavor, add more of the aioli. Salt to taste.
Spoon the saffron-egg yolk mixture into the egg white halves, and garnish with remaining chives.