Bring a medium pot of water to a boil, adding salt until it tastes like seawater. Meanwhile, set a colander in a large bowl of ice water. Add the parsley to the boiling water and cook 1 minute, then remove with a slotted spoon. Immediately plunge the parsley into the ice bath and allow to cool, 3 to 5 minutes. Keep the water at a rolling boil.
Drain the parsley slightly and put it in a blender (reserve the ice water). Add the sugar and olive oil and blend until smooth. If the mixture is too thick, add a little water. Taste for seasoning. Transfer to a bowl and stir in the scallions; set aside.
Prepare the asparagus: Add the asparagus to the pot and cook until crisp-tender and bright green, about 2 minutes. Transfer with a slotted spoon to the ice water to cool. Drain and pat dry. (The pesto and asparagus can be made up to 4 hours ahead; cover and refrigerate.)
Cook the scallops: Heat a large skillet over medium-high heat and add 2 tablespoons olive oil. When the oil begins to smoke slightly, season the scallops with salt and pepper and add to the pan. Cook until browned, 2 to 3 minutes per side. Remove from the pan and set aside.
Add the remaining 1 tablespoon olive oil to the pan. Add the mushrooms and cook until tender, 4 to 5 minutes. Season with salt and pepper. Reduce the heat to medium, then add the shallots and garlic and cook until translucent, 2 to 3 minutes. Add the vermouth and cook an additional 2 to 3 minutes, or until the vegetables are tender and virtually all of the liquid has evaporated. Add the asparagus and toss until warmed through.
Arrange the scallops and vegetables on plates. Squeeze some lemon juice directly over the scallops. Top with the pesto.