  Integrable sl(N) spin chains, which we consider in this paper, are not only
the prototypical example of quantum integrable systems but also systems with a
wide range of applications. For these models we use the Functional Separation
of Variables (FSoV) technique with a new tool called Character Projection to
compute all matrix elements of a complete set of operators, which we call
principal operators, in the basis diagonalising the tower of conserved charges
as determinants in Q-functions. Building up on these results we then derive
similar determinant forms for the form-factors of combinations of multiple
principal operators between arbitrary factorizable states, which include, in
particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We
prove that the set of principal operators generates the complete spin chain
Yangian. Furthermore, we derive the representation of these operators in the
SoV bases allowing one to compute correlation functions with an arbitrary
number of principal operators. Finally, we show that the available combinations
of multiple insertions includes Sklyanin`s SoV B operator. As a result, we are
able to derive the B operator for sl(N) spin chains using a minimal set of
ingredients, namely the FSoV method and the structure of the SoV basis.
