This study develops a Gâteaux differential-based mixed finite element formulation for bi-directionally functionally graded (BDFGM) Timoshenko beams graded along both the length and the thickness. The local effective properties are evaluated by the Mori–Tanaka scheme using two independent gradation indices, n_x and n_z, so that the section stiffnesses vary continuously along the beam axis. The axial strain is condensed under the zero-axial-force condition, and extension–bending coupling is retained through a spatially varying condensed bending stiffness. A two-node element with four independent fields (transverse displacement, rotation, bending moment, and shear force) is derived, and its variable bending and shear compliances are integrated by Gauss–Legendre quadrature. In the through-thickness limit (n_x = 0), using the rule of mixtures of the reference solutions, the present deflections agree with refined-theory benchmarks to within about 0.1%; mesh and quadrature studies confirm convergence for the bi-directional case. For n_z = 1 and k_s = 5/6, increasing n_x from 0 to 5 increases the maximum nondimensional deflection by about 59%, from 0.69465 to 1.10563 for L/h = 4 and from 0.61036 to 0.97469 for L/h = 16. For n_x = n_z = 1, replacing k_s = 1 by 5/6 increases the deflection by about 2.3% for L/h = 4 but by only 0.16% for L/h = 16.