Heat a large 10’’ skillet over high heat. Once very hot, add 2 Tbsp neutral oil, then add half the mushrooms. Toss to coat in the oil and make sure all of the pieces have surface contact with the pan, then lower the heat to medium-high, season with salt and pepper, and let sit, undisturbed, for 1-2 minutes. Stir the mushrooms and continue cooking until shrunken in size and browned, 8-10 minutes. Add 1/4 cup of white wine, scrape any brown bits from the bottom, 1 minute, then stir in 1 Tbsp butter. Transfer to a bowl and repeat for the second batch.
NOTE: If you notice the mushrooms are darkening too quickly, reduce the heat.
Transfer all mushrooms and brown bits to a bowl.
Heat the pan to medium-low, add the olive oil, garlic, shallots, and black pepper. Sauté until translucent, 10 minutes, then stir in the orzo and sauté, 2 minutes more. Add the vegetable stock, bring to a boil, then lower heat and simmer. Cover with a lid and simmer 10 minutes.
Stir in the mushrooms and Parmesan cheese until warmed and melted.
Serve warm with dollops of ricotta and fresh thyme.