Heat oven to 425 degrees. Cut 4 pieces of parchment paper, 12 by 16 inches each. Divide brown rice among each parchment piece. Top with mushrooms and thyme. Season with salt and pepper and drizzle with olive oil.
Bake on a rimmed baking sheet until packet is puffed and mushrooms are cooked through, 20 to 25 minutes. Serve with greens divided among plates and squeezed with lemon wedges.