Foil Stamping
Foil stamping is the process of applying a logo using a magnesium plate and foil transferred via heat and pressure. It is an excellent option for 1-colour logos, or anything where a metallic shade is required. Because much of the cost of foil stamping is an upfront cost for the plate, this is not the best choice small quantities of menu covers, unless you anticipate reordering them regularly without any changes to the logo. This process is also used for debossing (pressing in) the logo, so the same caveats apply.
Foil colours available run the gamut, though metallic shades tend to offer a much crisper result than pigment shades. White is particularly difficult and it might be worth considering UV print as an alternative if you require a crisp white logo.
For materials, boot leather, buckram and faux leather stamp quite well. Linens, due to their open weave and lack of coating present issues.