When complex mechanisms are involved, pinpointing high-performance materials within large databases is a major challenge in materials discovery. We focus here on phonon-limited conductivities, and study 2D semiconductors doped by field effects. Using state-of-the-art density-functional perturbation theory and Boltzmann transport equation, we discuss 11 monolayers with outstanding transport properties. These materials are selected from a computational database of exfoliable materials providing monolayers that are dynamically stable and that do not have more than 6 atoms per unit cell. We first analyze electron-phonon scattering in two well-known systems: electron-doped InSe and hole-doped phosphorene. Both are single-valley systems with weak electron-phonon interactions, but they represent two distinct pathways to fast transport: a steep and deep isotropic valley for the former and strongly anisotropic electron-phonon physics for the latter. We identify similar features in the database and compute the conductivities of the relevant monolayers. This process yields several high-conductivity materials, some of them only very recently emerging in the literature (GaSe, Bi₂SeTe₂, Bi₂Se₃, Sb₂SeTe₂), others never discussed in this context (AlLiTe₂, BiClTe, ClGaTe, AuI). Comparing these 11 monolayers in detail, we discuss how the strength and angular dependency of the electron-phonon scattering drives key differences in the transport performance of materials despite similar valley structure. We also discuss the high conductivity of hole-doped WSe₂, and how this case study shows the limitations of a selection process that would be based on band properties alone. In this entry we provide the AiiDA database with the calculations of phonons and electron-phonon interactions for the 11 materials, along with the python library to collect and visualise the data, solve the Botzmann transport equation, and launch the same workflows for other 2D materials. To guide the reader, we include a Jupyter notebook showing how to extract the data, use the basic functionalities of the library, and regenerate the plots included in the associated paper.