Preheat oven to 425 degrees.
Add potatoes to a large rimmed baking sheet. Drizzle with three tablespoons olive oil and sprinkle with half teaspoon salt and black pepper. Use hands to evenly coat the potatoes with oil, salt, and pepper. Spread evenly on the baking sheet. Pop in the oven and set the timer for 20 minutes.
After you put the potatoes in the oven, add mushrooms to another large rimmed baking sheet. Drizzle with remaining 3 tablespoons olive oil, sprinkle with 1/2 teaspoon salt. Use hands to evenly coat the mushrooms with the oil and salt. Spread evenly on the baking sheet and place 4 thyme sprigs on top. Place in the oven with the potatoes.
When the 20 minutes are up, pull both the rack with the potatoes out. Take the mushrooms out of the oven completely, they should be golden brown and crispy at this point. If they aren't, put them back in for 2-3 minutes. Use a spatula to toss the potatoes Sprinkle the potatoes with 4 sprigs of thyme and 1/3 cup of parmesan cheese. Toss one more time. Place back in the oven and roast until cheese is melted and potatoes are golden brown and crisp, another 10-15 minutes.
Sprinkle the hot mushrooms with 3 tablespoons parmesan. Toss one time, set aside until potatoes are done.
When the potatoes are done, add the mushrooms to the same pan as the potatoes and toss everything together. Remove the thyme stems.
Garnish with fresh thyme leaves. If desired, squeeze a little lemon juice on top.