  Let $p$ be a prime number and $Gal_p$ be the Galois group of the maximal
abelian extension unramified outside $p\infty$ of a totally real number field
$F$ of degree $d$. Using automorphic cycles, we construct evaluation maps on
the top degree $t =d (n^2 + n -1)$ parahoric overconvergent cohomology for
$GL_{2n}/F$, which allow us to attach a distribution on $Gal_p$ of controlled
growth to any finite slope overconvergent $U_p$-eigenclass. When the eigenclass
comes from a non-critical refinement $\tilde\pi$ of a cuspidal automorphic
representation $\pi$ of $GL_{2n}(\mathbb{A}_F)$ which is spherical at $p$ and
admits a Shalika model, we prove this distribution interpolates all
Deligne-critical $L$-values of $\pi$, giving the first construction of $p$-adic
$L$-functions in this generality beyond the $p$-ordinary setting. Further,
under some mild assumptions we use our evaluation maps to show that $\tilde\pi$
belongs to a unique $(d+1)$-dimensional Shalika family in a parabolic
eigenvariety for $GL_{2n}/F$, and that this family is \'etale over the weight
space. Finally, under a hypothesis on the local Shalika models at bad places
which is empty for $\pi$ of level 1, we construct a $p$-adic $L$-function for
the family.
