Every smooth even asymptotically hyperbolic space \(\bar{M}\) with boundary can be doubled through a natural procedure, producing a smooth conformal manifold without boundary. In the case of a Poincaré-Einstein metric, this doubling leads to an Einstein manifold with hypersurface singularity. In the present article, the author generalizes this construction to what is called a collapsing sphere product \(D_\ell \bar{M}\) or \(S^\ell\)-doubling and investigates geometric properties of this construction when applied to even asymptotically hyperbolic Einstein spaces. Roughly, the space \(D_\ell \bar{M}\) is obtained by identifying at each boundary point of \(\bar{M}\) the sphere \(S^\ell\) in the product \(S^\ell \times \bar{M}\) with a single point. The collapsing sphere product \(D_\ell H^{n+1}\) of the Poincaré model of hyperbolic space \(H^{n+1}\) is conformally equivalent to the Möbius sphere of the corresponding dimension. The author shows that the \(S^\ell\)-doubling of an even asymptotically hyperbolic Einstein space admits multiple almost Einstein structures with intersecting scale singularities and decomposable conformal holonomy. The pole of \(D_\ell\bar{M}\) is shown to be a totally umbilical submanifold. The author also presents explicit constructions of a Poincaré-Einstein metric and a Fefferman-Graham ambient metric for \(D_\ell\bar{M}\).