Make the dressing: Combine the lemon juice, zest, thyme, garlic, and sugar in a blender or food processor; blend until smooth. With the machine running, add the oil in a slow, steady stream. Season with salt and pepper.

Prepare the salad: Heat the grill. Toss the asparagus and mushrooms with the oil and sprinkle with salt. Put the mushrooms on a medium-high part of the grill; turn them occasionally to prevent sticking. After about 10 minutes, add the asparagus and grill it, rolling frequently to avoid burning. Grill the mushrooms until dark and meaty-looking, about 15 minutes total; they should be crisp at the edges but still juicy. Cook the asparagus until browned and crisp-tender, 5 to 7 minutes, depending on thickness. Line four salad plates with the greens. Slice the mushrooms and arrange them and the asparagus on the greens. Drizzle with the dressing and serve.