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Put the eggs in a single layer in a pan and add enough water to cover them by 2cm. Bring to the boil, then cover, turn off the heat and leave for 12 mins. Drain, cool completely in very cold water, then drain again.
Put the different food colourings into separate bowls or containers that will hold the eggs, using 1 tsp food colouring per bowl, plus 700ml water and 1 tbsp of the vinegar each. Gently tap the eggs all over using a spoon or knife to crack the shells without removing them. Divide the cracked eggs between the bowls with the food colourings, submerge them in the liquid and leave to soak for at least 3 hrs, or ideally overnight. If you can’t submerge them fully, turn the eggs every hour so that they stain evenly.
Drain, rinse and peel the eggs, then halve them and pop the yolks into a bowl. Add the mayonnaise, adding a little more if you need to in order to create a slightly stiff paste, then mix in 3 tbsp of the chopped pepper and the tarragon, if using. Spoon the yolk mixture back into the whites, then sprinkle over the remaining pepper and the paprika.