  The J=1/2 iso-doublet double-$\Lambda$-hypernuclei, namely,
${}_{\Lambda\Lambda}^{5}$H and ${}_{\Lambda\Lambda}^{5}$He, are examined as the
three-body cluster states, $\Lambda\Lambda t$ ($t\equiv {}^3$H or triton) and
$\Lambda\Lambda h$ ($h\equiv {}^3$He or helion), respectively, in a model
independent framework utilizing pionless halo effective theory. Both singlet
and triplet states of the constituent $\Lambda T$ ($T\equiv t,h$) sub-system
are used in the elastic channel for the study of $_\Lambda^4$H$-\Lambda$ and
$_\Lambda^4$He$-\Lambda$ scattering processes. A prototypical leading order
investigation using a sharp momentum cut-off regulator, irrespective of the
type of the elastic channel chosen, yields almost identical cut-off dependence
of the three-body binding energy or the double-$\Lambda$-separation energy
($B_{\Lambda\Lambda}$) is obtained for the mirror partners, evidently
suggesting good isospin symmetry in these three-body systems. Subsequently,
upon normalization of our solutions to the integral equation with respect to a
single pair of input data from an {\it ab initio} potential model analysis for
each mirror hypernuclei, yields $B_{\Lambda\Lambda}$ which agrees fairly well
with various erstwhile regulator independent potential models for our choice of
the cut-off regulator, $\sim 200$ MeV. This is either consistent with pionless
effective theory or with its slightly augmented version with a hard scale
around $2m_\pi$, where low-energy $\Lambda$-$\Lambda$ interactions dominated by
$\pi\pi$ or $\sigma$-meson exchange. Finally, to demonstrate the predictability
of our effective theory, we present preliminary estimates of the S-wave
$\Lambda\Lambda T$ three-body scattering lengths and the $\Lambda$-separation
energies.
