Step 1Trim the stalks and fronds from the fennel, reserving enough fronds to make 1/4 cup when chopped. Cut the bulb in quarters lengthwise and trim away the triangle of solid core at the base. Cut each quarter into thirds or quarters lengthwise and cut each of those in half crosswise.
Step 2Trim the tough bottoms of the mushroom stems and cut the mushrooms into quarter-inch slices.
Step 3In a large bowl, whisk together the olive oil, lemon juice, green onions, thyme leaves, the fennel fronds, salt and garlic, to mix well. Add the fennel and stir to coat. Remove the fennel with a slotted spoon, draining any excess dressing back into the bowl, and arrange the fennel on a platter in a broad, shallow layer.
Step 4Add the mushrooms to the leftover dressing, season to taste with black pepper and stir to coat well. Arrange the mushrooms in an oblong mound on top of the fennel, centering it so there is fennel showing around the edges.
Step 5Using a vegetable peeler, shave large flakes of Parmesan over the top of the salad. Serve immediately.