Warm olive oil in a large skillet over medium heat.
Cook onion and garlic until softened and fragrant.
Add mushrooms and cook until browned, about 6 minutes.
Stir in orzo and toast lightly for 1–2 minutes.
Pour in broth and milk, bring to a simmer, and cook until orzo is tender, about 10 minutes, stirring often.
Fold in spinach until wilted.
Stir in Parmesan, season with salt and pepper, and serve warm.