Defying the traditional classification into metals and insulators, several materials simultaneously display metallic thermal properties and insulating electric behaviour, as if they hosted quasiparticles carrying entropy but not charge. Among them, some materials also possess quantum oscillations in magnetic fields as if they had well-defined Fermi surfaces despite the insulating gap. This remarkable dichotomy has been observed in the topological Kondo insulators SmB$_6$ and YbB$_{12}$. Prompted by the peculiar mixed-valence nature of these compounds, involving $f$ and $d$ electrons of the lanthanide, we propose an explanation of their intriguing properties drawing inspiration from the physics of the pseudogap phase in underdoped cuprates. We argue that the $f$ and $d$ subsystems, when considered separately, act, respectively, as electron- and hole-doped Mott insulators, featuring Fermi pockets coexisting with Luttinger surfaces responsible for the pseudogap. When the two are coupled to each other a hybridisation gap opens up, and the whole turns into a topological insulator endowed with genuine chiral edge states. However, the Luttinger surfaces persist and support neutral quasiparticles. This scenario, supported by numerical simulations within the dynamical cluster approximation, effectively resolves the paradoxical phenomenology of SmB$_6$ and YbB$_{12}$.