Make the crepes: In a large bowl, whisk together eggs and salt. Gradually whisk in flour to create a paste, then little by little the milk. Strain into a medium bowl. Let stand 1 hour.
If necessary, add more milk, a tablespoon at a time, until the batter reaches the consistency of heavy whipping cream.
Heat a nonstick skillet over medium-high heat. Brush with butter, then pour 4 tablespoons of batter into skillet, swirling the pan to coat the bottom evenly. Fully cook until golden on one side, then flip and cook 30 seconds more. Remove to a plate and continue with remaining batter.
Make the filling: Heat oil in a large skillet over high heat, then add in mushrooms and cook until reduced in size, stirring occasionally, about 5 minutes. Add in shallots, garlic, thyme, rosemary and butter and stir to combine, about 30 seconds. Season with salt and pepper, and squeeze in lemon juice. Add spinach and let wilt. Stir to mix, then remove from heat.
Assemble the crepes: Preheat oven to 425 degrees, and place the rack in the lowest row of the oven.
On a clean work surface, overlay two crepes by 3 inches, then spoon 3 large tablespoons of mushroom mixture in a row in the center of the crepes. Crumble 2 tablespoons of goat cheese on top of the filling. Beat an egg in a small bowl, then brush all corners of the crepes with the egg wash. Gently roll the crepes into a log, then roll the log into a coil shape. Repeat with the remaining crepes and filling.
Place rolls in a 10-inch cast-iron pan and brush with egg wash. Sprinkle generously with Parmesan. Gently make a divot in the center of each crepe roll with the back of a spoon. Crack an egg into each divot. Bake for 10 to 15 minutes or until the egg whites are opaque but the yolks are still runny. The crepes should be crispy and golden on the outside. Sprinkle with chopped chives and salt and serve immediately.