  A systematic development of the so-called Palatini formalism is carried out
for pseudo-Finsler metrics $L$ of any signature. Substituting in the classical
Einstein-Hilbert-Palatini functional the scalar curvature by the Finslerian
Ricci scalar constructed with an independent nonlinear connection $\mathrm{N}$,
the affine and metric equations for $(\mathrm{N},L)$ are obtained. In
Lorentzian signature with vanishing mean Landsberg tensor Lan$_i$, both the
Finslerian Hilbert metric equation and the classical Palatini conclusions are
recovered by means of a combination of techniques involving the (Riemannian)
maximum principle and an original argument about divisibility and fiberwise
analyticity. Some of these findings are also extended to (positive definite)
Riemannian metrics by using the eigenvalues of the Laplacian. When Lan$_i\neq
0$, the Palatini conclusions fail necessarily, however, a good number of
properties of the solutions remain. The framework and proofs are built up in
detail.
