In a medium bowl, add the asparagus and 1/2 tablespoon olive oil. Season with salt and pepper and gently toss to combine. Heat 1/2 tablespoon olive oil in a large skillet over high heat. Add the asparagus and cook on both sides, 4 minutes total. Remove from the heat and set aside.
Add water to a small pot and bring to a low boil. Add the vinegar. Crack the egg and gently add to the pot. Cook for 2-3 minutes. Remove from the pot and set aside.
Toast the bread. Drizzle a little bit of olive oil over the bread and arrange the sorrel on top. Add the asparagus, egg, anchovies, and drizzle with additional olive oil. Sprinkle with the chopped sorrel and a pinch of sea salt and serve immediately with a lemon wedge on the side.