Heat 1 tablespoon butter in a large skillet over medium-high heat. Add leftover orzo and baby spinach; season with salt and pepper. Cook, stirring, until orzo is hot and spinach is wilted, 4 to 5 minutes. Remove from skillet; keep warm.
Heat remaining tablespoon butter in same skillet over medium heat. Fry eggs in skillet until edges are firm, about 1 minute; partially cover, and cook to desired doneness. Serve eggs on top of orzo hash.