Rinse the rice under cold water until mostly clear; drain very well.
Warm butter and olive oil in a medium pot over medium heat.
Add mushrooms and cook, stirring occasionally, until browned and reduced, 6–8 minutes.
Stir in onion and cook until translucent, about 3 minutes; add garlic and cook 30 seconds.
Season with soy sauce, thyme, salt, and pepper; stir to coat the vegetables.
Add drained rice and toast, stirring, for 1–2 minutes until glossy.
Pour in broth and bring to a lively simmer; reduce heat to low, cover, and cook 15 minutes.
Stir in peas, if using, cover again, and cook 3–5 minutes more until rice is tender.
Remove from heat and rest, covered, 5 minutes. Fluff with a fork; fold in parsley and lemon juice. Sprinkle Parmesan to serve.