  We propose a way to separate variables in a rational integrable
$\mathfrak{gl}(n)$ spin chain with an arbitrary finite-dimensional irreducible
representation at each site and with generic twisted periodic boundary
conditions. Firstly, we construct a basis that diagonalises a higher-rank
version of the Sklyanin B-operator; the construction is based on recursive
usage of an embedding of a $\mathfrak{gl}(k)$ spin chain into a
$\mathfrak{gl}(k+1)$ spin chain which is induced from a Yangian homomorphism
and controlled by dual diagonals of Gelfand-Tsetlin patterns. Then, we show
that the same basis can be equivalently constructed by action of
Backlund-transformed fused transfer matricies, whence the Bethe wave functions
factorise into a product of ascending Slater determinants in Baxter
Q-functions. Finally, we construct raising and lowering operators -- the
conjugate momenta -- as normal-ordered Wronskian expressions in Baxter
Q-operators evaluated at zeros of B -- the separated variables. It is an
immediate consequence of the proposed construction that the Bethe algebra
comprises the maximal possible number of mutually commuting charges -- a
necessary property for Bethe equations to be complete.
