Melt 3 Tbsp butter in 10" cast iron pan at 400°
While the pan is warming, make the pastry. Cut in flour to shortening with a pastry blender or fork until crumbs form.
Once the pan is good and hot and butter is melted toss in chopped asparagus, coat in butter, and return to oven for 7 minutes to cook
Continue with pastry. In a separate bowl, beat the egg and stir in the salt, vinegar, and half the cold water. Cut egg mixture into flour mixture until well combined. Add extra ice water slowly if needed to make a smooth dough ball that is not sticky or crumbly
Roll out dough on a well-floured surface until it is about 3" larger in diameter than your cast iron cooking pan. I used a 12" pan. Transfer dough to pan and coat the bottom of the dough with a little of the beaten egg white from the separated egg
For the filling: stir together the milk, flour, and seasonings
Divide one egg yolk and white, reserving both parts. Add 4 whole eggs and the extra yolk to milk mixture and stir well
Pull asparagus from oven and sprinkle over the pie crust, and top with mozzarella cheese
Pour beaten egg mixture evenly over the asparagus
Top with 3/4 cup Dubliner cheese and fold pie crust edges in over the egg mixture. Brush with beaten egg white and remaining tbsp cold water
Bake in 375° oven for approx 20 minutes, or until set
Cook the remaining whole spears in the former buttered asparagus pan during the last 8 minutes of baking time
Remove galette from the oven and immediately top with asparagus spears and remaining 1/4 cup Dubliner cheese. Serve warm!