Two-dimensional nanomaterials play a critical role in biology (e.g., lipid bilayers) and electronics (e.g., graphene) but are difficult to directly synthesize with a high level Work of precision. Peptoid nanosheet bilayers are a versatile synthetic platform for constructing multifunctional, precisely ordered two-dimensional nanostructures. Here we show that nanosheet formation occurs through an unusual monolayer intermediate at the air-water interface. Lateral compression of a self-assembled peptoid monolayer beyond a critical collapse pressure results in the irreversible production of nanosheets. An unusual thermodynamic cycle is employed on a preparative scale, where mechanical energy is used to buckle an intermediate monolayer into a more stable nanosheet. Detailed physical studies of the monolayer-compression mechanism revealed a simple preparative technique to produce nanosheets in 95% overall yield by cyclical monolayer compressions in a rotating closed vial. Compression of monolayers into stable, free-floating products may be a general and preparative approach to access 2D nanomaterials.