Heat butter in nonstick skillet over medium-high heat; when foaming subsides, add onion and salt and cook, stirring frequently, until onion has softened and is beginning to brown, about 5 minutes. Add garlic and cook until fragrant, about 30 seconds. Add orzo and cook, stirring frequently with heatproof rubber spatula, until most of orzo is lightly browned and golden, 5 to 6 minutes.

Off heat, add vermouth and chicken broth. Return skillet to medium-high heat and bring to boil; reduce heat to medium-low and simmer, stirring occasionally, until all liquid has been absorbed and orzo is tender, 10 to 15 minutes.

With rubber spatula, stir in peas, Parmesan, nutmeg, and pepper to taste. Let stand off heat until peas are heated through, about 2 minutes; adjust seasoning with salt and serve.