When the mixture is bubbly and active, add the rest of the flour, the sugar, the honey, the salt, the spices, the olive oil and the two eggs. Break the third egg into a cup, beat it with a fork and add half of it to the bowl with the rest of the ingredients. Reserve the other half for the egg wash.
Let the dough rest in a warm place until tripled in volume (a closed oven with only the oven-light on, should do the trick). It may take 2-5 hours depending on the yeast and the rest of the conditions. Only when the dough has tripled in volume, you will oil your hands and knead it once again on your counter. If it is still too sticky, use the extra flour.
Preheat your oven to 356°F (180°C).
Beat the reserved half egg with 1 tablespoon milk, 1 teaspoon sugar, and the vanilla and gently brush the surface of the bread. Then, sprinkle it with some sliced almonds. Bake for 30-40 minutes. If you notice the surface getting browned too quickly, cover it with a piece of foil.
Take it out of the oven and let it rest on a rack to cool. Eat!
Tsoureki should be stringy and moist inside (but not underbaked). Having said that, take care not to overbake it because it can dry out easilly.