We show that the production of $J/\psi$ or ϒ pairs in unpolarised pp collisions is currently the best process to measure the momentum distribution of linearly-polarised gluons inside unpolarised protons through the study of azimuthal asymmetries. Not only the short-distance coefficients for such reactions induce the largest possible $\cos 4\varphi$ modulations, but analysed data are already available. Among the various final states previously studied in unpolarised pp collisions within the TMD approach, di-$J/\psi$ production exhibits by far the largest asymmetries, in the region studied by the ATLAS and CMS experiments. In addition, we use the very recent LHCb data at 13 TeV to perform the first fit of the unpolarised transverse-momentum-dependent gluon distribution.