We report on the realization of ultrasensitive absolute pressure sensors based on silicon nitride membrane sandwiches. These sandwiches consist in a pair of highly-pretensioned, ultrathin (50 nm), large area (0.25 mm2) films, suspended parallel to each other and forming an ultrashort (500 nm), open cavity. The compression of a gas in this cavity leads to a strong squeeze film force, resulting in an increase in the membrane mechanical resonance frequencies which is directly proportional to the absolute gas pressure. These sandwiches show a record high responsitivity of >300 Hz/Pa in terms of squeeze film-induced frequency shift, which, combined with high quality factor mechanical resonances (Q>10^6), allows for bringing the sensitivity of absolute squeeze film pressure sensors down to the sub-millipascal level.