Trim the asparagus (keeping the darker green and less fibrous parts) and cut into 1-inch pieces.
Thinly slice the mushrooms.
Simmer the asparagus in 1/2 cup of water with a pinch of salt until it is tender (about 10-13 minutes).
While the asparagus is simmering, whisk the lemon juice, vinegar, Dijon mustard, and garlic together.
Gradually add the olive oil and then the black pepper.
Once you have drained the asparagus and let it cool, gently toss it with half of the dressing to coat it well.
Toss the remaining dressing with the basil and the mushrooms, and sprinkle the mixture over the asparagus.
Season to taste with salt, garnish with freshly grated Parmesan or Mizithra cheese, and serve.
