For the crepes: Add the eggs, milk, club soda and a pinch of salt to a blender and process until completely combined. Pour in the 3 tablespoons melted butter, then add the flour and blend until smooth. The mixture should be like VERY loose pancake batter. If the mixture is a little thick, add a little more milk. Let the batter sit for at least 30 minutes before using.
Melt about 1/2 teaspoon butter in a small nonstick saute pan over medium heat. The butter should coat the bottom of the pan; if there is a lot of excess, wipe it out with a paper towel. Keep the paper towel handy in case you need it again (you will).
Fill a 2-ounce ladle almost to the top with batter and pour it into the preheated pan, tipping and rolling the pan as you ladle the batter to evenly cover the bottom. This will take a little practice; even when you are an experienced crepe maker the first couple always get wasted. Accept it and move on.
When the edges of the crepe begin to pull away from the edges of the pan and the bottom begins to brown a little, turn the crepe over and cook it for about 1 minute on the other side. Remove the crepe from the pan to a plate and let cool. Stack the crepes as they are cooked between parchment paper squares. Repeat this process until all the batter is used, wiping the pan with your paper towel or melting a little more butter in the pan as needed.
For the ham and Brie crepes: Preheat the oven to 300 degrees F.
Divide the ham and cheese evenly among 8 crepes (save the remaining crepes for another use). Roll each crepe and place on a sheet pan fitted with a rack. Bake to melt the cheese and warm the ham, 6 to 8 minutes.
Heat a nonstick skillet over medium-high heat and add olive oil to coat. Crack the eggs in the skillet (you can do this in batches if you prefer). Cook until the whites are just set, a minute or two. Remove from the heat and set aside.
Whisk to combine the shallots, vinegar, Dijon, and 1/2 cup olive oil in a medium bowl. Dress the greens lightly and set aside.
Plate 2 crepes on each of 4 serving plates and top each with 1 egg and a handful of the dressed greens. Enjoy!