Preheat the oven to 400 degrees F (200 degrees C).
In a food processor, add in the All-Purpose Flour (1 1/4 cup) and Salt (1/4 teaspoon) and pulse twice to combine. Add in the Unsalted Butter (8 tablespoon) and pulse until the mixture resembles a coarse meal.
Mix the Greek Yogurt (1/4 cup), Water (1/4 cup), and Lemon Juice (2 teaspoon) in a medium mixing bowl, and then mix in the mixture from the food processor. On a clean floured surface, use your hands to form a ball. Wrap in plastic wrap and put in the freezer to chill for 20 minutes.
Cut off the ends of the Asparagus (450 gram) and using a peeler, peel the base to remove the outer layer and make them uniform in size.
In a large mixing bowl, mix together the Feta Cheese (1/2 cup) and Parmesan Cheese (1/2 cup) using a fork. Then add in Olive Oil (1 tablespoon), Ground Black Pepper (1 teaspoon), and Garlic (1 clove), mix until well combined. Put in the fridge until ready to use.
On a well-floured surface, roll the dough into a 10x15-inch rectangle. Transfer the dough onto a parchment-lined baking sheet by rolling it and then unrolling it along the rolling pin.
Push the cheese mixture onto the dough, then spread carefully and evenly, leave a 2-inch border on the edges.
Arrange the asparagus on top of the cheese mixture and spread the Cherry Tomato (8) evenly over the top. Drizzle some more olive oil on top. Fold in the edges so the four sides look smooth. Add a sprinkle of salt onto the tomatoes.
Brush the edges with Egg (1) wash. Put in the oven and bake for about 25 minutes, or until the edges are slightly golden brown.
Take out the galette, crack the Egg (3) on top, then put back into the oven and bake for another 5-8 minutes, until the egg white is set.
Garnish with Fresh Parsley (to taste) and Feta Cheese (to taste). Add salt and pepper to taste. Serve hot.