  The goal of this work is to find the simplest UV completions of Accidental
Composite Dark Matter models that can dynamically generate an asymmetry for the
Dark Matter candidate, the lightest dark baryon, and simultaneously annihilate
the symmetric component. In this framework Dark Matter is an accidentally
stable bound state of a confining $\mathrm{SU}(N)$ gauge group that can
interact weakly with the visible sector. The generation of asymmetry for such
candidate happens via the out-of-equilibrium decay of two flavors of a heavy
scalar $\phi$, with mass $M_\phi > 10^{10}$ GeV. Below such scale, the models
recover accidental stability, or long-livedness, of the Dark Matter candidate.
The symmetric component is annihilated by residual confined interactions
provided that the mass of the dark baryon is < 75 TeV.
