Set up your Big Green Egg for raised direct cooking. Preheat the dome temperature to 400°F.
Clean and trim the mushrooms. Cut any large mushrooms into bite-sized pieces for even cooking.
Place the mushrooms in a large bowl. Drizzle the mushrooms with 2 tablespoons olive oil. Season with kosher salt and black pepper. Toss until evenly coated.
Tear the sourdough loaf into rustic bite-sized chunks. Lightly drizzle the bread with the remaining 1 tablespoon olive oil.
Place the mushrooms in a grill basket, perforated pan, or cast iron skillet. Place the mushrooms on the grill over raised direct heat.
Grill the mushrooms for 15–20 minutes, stirring occasionally, until tender and caramelized.
Place the bread chunks directly on the grill grate. Grill the bread for 3–5 minutes, turning as needed, until crisp and lightly charred.
Transfer the grilled mushrooms and bread to a large mixing bowl while still warm.
Add the minced garlic. Add the fresh lemon juice. Add the chopped parsley. Add the microgreens. Add 3/4 cup of the grated parmesan cheese.
Toss gently to combine. Taste and adjust with extra olive oil, lemon juice, salt, or pepper as needed.
Transfer to a serving platter or bowl. Garnish with the remaining parmesan cheese.
Serve warm or slightly cooled.