Remove flank steak from fridge 30 to 60 minutes ahead of preparation.

Preheat oven to 425 F. Line medium baking sheet with parchment paper.

For mushroom mixture, to large skillet on medium heat, add olive oil and garlic and sauté for 1 minute. Add mushrooms, rosemary, thyme, and 1/2 tsp sea salt; sauté approximately 10 minutes, or until mushrooms are fragrant and soft. Turn off heat and set aside.

For cauliflower purée, to prepared baking sheet, add carrots, cauliflower, onion, garlic, olive oil, sea salt, and black pepper; mix. Roast in preheated oven for 15 to 20 minutes, or until carrots are fork tender, turning vegetables at 8 minutes. Remove from oven and set aside. Reduce oven temperature to 350 F.

Transfer roasted vegetables to high-speed blender, add vegetable broth and oat milk, and blend until smooth.

Line large cutting board with parchment paper.

With sharp knife, carefully butterfly flank steak by cutting it horizontally almost all the way through. Open steak like a book and place on cutting board. Cover steak with parchment paper. Using meat mallet or rolling pin, pound steak into an even thickness of 1/4 to 1/2 inch.

Spread even layer of Parmigiano Reggiano over steak surface, followed by parsley and mushroom mixture. Tightly roll steak from the small end first in direction of meat grain, keeping mushroom mixture inside as you roll tightly. Secure well with kitchen twine or bamboo toothpicks every 2 inches. Season exterior of steak with 1/4 tsp pepper and remaining 1/4 tsp sea salt. With sharp knife, cut rolled steak between kitchen twine or bamboo toothpicks, making 2 inch pinwheels.

In cast iron skillet on medium to high heat, add avocado oil and sear pinwheels for 2 minutes on each side. Place cast iron skillet with pinwheels into preheated oven for 15 minutes, until steak is medium rare with an internal temperature of 135 F (see tip) Remove cast iron skillet from oven and let pinwheels rest for 10 minutes before serving.

When plating, spoon steak juices from pan over pinwheels and serve with roasted carrot and cauliflower purée.