For the tsoureki, first prepare the eggs. To do this, hard boil the eggs in a saucepan with water for 10 minutes.
Mix the red food coloring, vinegar and water in a bowl. Remove the eggs from the water, blot them briefly and then roll them in the color. Cool on a broiler and dry. Drizzle a piece of kitchen roll with the oil and use it to shine the cooled eggs, one at a time.
Lightly grease a baking tray. For the batter, sift salt, flour, new spice and cinnamon into a large baking bowl. Stir in the caraway seeds.
Stir the yeast with the milk until smooth. Cream butter and sugar in a baking bowl, mix in eggs. Add the mixture to the flour together with the yeast mixture and knead into a dough. Knead the dough on the surface until it is smooth and elastic.
Place the dough in a lightly oiled baking bowl, cover with lightly oiled plastic wrap and let rise in a warm place for about 1 hour.
Then remove the dough and knead for 2 to 3 minutes. Return to the baking bowl, cover and let rise repeatedly in a warm place for about 1 hour.
Place dough on floured work surface and divide into thirds. Shape each third into a 40- to 50-inch roll and braid, starting from the center and working toward the ends.
Place the braid on the prepared baking sheet and press the dyed eggs into it. Cover and let rise in a warm place for another 1 hour.
In the meantime, set the oven to 190 °C (level 3).