  Energy transport in quantum many-body systems with well defined
quasiparticles has recently attracted interest across different fields,
including out of equilibrium conformal field theories, one dimensional quantum
lattice models and holographic matter. Here we study energy transport between
\emph{strange quantum baths} without quasiparticles, made by two
Sachdev-Ye-Kitaev (SYK) models at temperatures $T_L\neq T_R$ and connected by a
Fermi-Liquid system. We obtain an exact expression for the nonequilibrium
energy current, valid in the limit of large bath and system size and for any
system-bath coupling $V$. We show that the peculiar criticality of the SYK
baths has direct consequences on the thermal conductance, which above a
temperature $T^*(V)\sim V^4$ is parametrically enhanced with respect to the
linear-$T$ behavior expected in systems with quasiparticles. Interestingly,
below $T^*(V)$ the linear thermal conductance behavior is restored, yet
transport is not due to quasiparticles. Rather the system gets strongly
renormalized by the strange bath and becomes Non-Fermi-Liquid and maximally
chaotic. Finally, we discuss the full nonequilibrium energy current and show
that its form is compatible with the structure
$\mathcal{J}=\Phi(T_L)-\Phi(T_R)$, with $\Phi(T)\sim T^{\gamma}$ and power law
crossing over from $\gamma=3/2$ to $\gamma=2$ below $T^*$.
