  In a recent letter we proposed a new non-thermal mechanism of Dark Matter
production based on vacuum misalignment, where both the Higgs boson and a very
light pseudo-scalar $\eta$ emerge from the Dark sector. In this letter, we
identify the parameter space in a composite scenario where the light
pseudo-scalar can be produced in the sun and explain the XENON1T excess in
electron recoil data. The model's Dark Matter candidate has a mass around $50$
TeV and out of range for Direct Detection. Testable predictions include
Gravitational waves at frequencies in the Hz range from a cosmological phase
transition, an exotic decay $Z \to \gamma + \mbox{inv.}$ with rates $4 \div 16
\cdot 10^{-12}$ testable at a future Tera-Z collider, and an enhancement by
$17\div 40$ % of the branching ratio $K_L \to \pi^0 + \mbox{inv.}$, not enough
to explain the KOTO anomaly. All these predictions may be confirmed by future
experiments.
