Heat the chicken stock in a saucepan over medium heat. Add the chopped carrots, celery, and onion and cook until the vegetables are soft, about 10 minutes.
Meanwhile, break the eggs into a medium bowl, add 1 teaspoon water, and beat lightly. Add the flour and whisk well to combine. (You're essentially making REALLY thin pancake batter.) Brush a crepe pan (or a small nonstick skillet) with a little olive oil and place it over medium-high heat. Pour in a small amount of batter and swirl it around to thinly coat the bottom of the pan; pour any excess back into the bowl. Cook until set, about 30 seconds to 1 minute. Carefully turn the crepe over and cook on the other side for 10 to 15 seconds. Remove the crepe and put it onto a plate. Continue making crepes and stacking them until all the batter is used up. When the crepes cool, slice the stack into 1/4-inch wide strips.
Heat 1 teaspoon olive oil in a small skillet over medium-high heat. Cook the mushrooms until they release their liquid, about 3 minutes; set aside. Add another teaspoon oil and cook the romaine until just wilted, about 1 minute; set aside.
Divide the soup among 4 heated wide soup bowls. Pile on the sliced crepes, then top with the romaine and mushrooms. Serve immediately.
The soup stock can be anything you like. Fritatten soup is a really nice base that you can take in any direction you like.