Preheat oven to 450F or 425F convection. Line a baking sheet with parchment. Add cauliflower, mushrooms, 2 tablespoons oil, thyme, and ½ teaspoon salt and toss to coat well. Spread out into an even layer and bake for 25 minutes or until cauliflower is crispy and charred and mushrooms are golden.
Fill a large pot with water (about 3 quarts) and bring to a boil over high heat. Salt the pasta water (about 1 tablespoon), then add the pasta and cook according to the package instructions. Reserve 1 cup of pasta water, then drain and transfer back to the pot.
Meanwhile, in a high-speed blender, add the walnuts and pulse to finely chop. Transfer about ¼ cup of the walnuts to a bowl to use for topping later.
In the blender with the walnuts, add the remaining ¼ cup olive oil, cheese, pepper, remaining ½ teaspoon salt, and about ½ cup of the reserved pasta water. Blend on high until smooth. Add more pasta water, if needed to create a smooth sauce.
Transfer the veggies to the pot with the pasta. Pour the sauce over top, scraping out every last bit, and stir to coat the noodles. If it’s looking dry, add more pasta water and stir again (the pasta will soak up the sauce as it sits - best to eat this right away!).
Serve topped with the reserved walnuts, parm, and pepper.