Fourier's law is just expressing the heat flux as a function of the gradient of temperature in a medium characterized by the heat conductivity $k$. What's the effect of such heat flux on the temperature of the material depends on something else which are not present in the Fourier's law: the effect of a heat transfer on the temperature rate of variation. Under the hypothesis that all the heat entering in a small volume of material goes into internal energy, one can obtain the Fourier equation of heat ( https://en.wikipedia.org/wiki/Heat_equation ), which in one dimension, and for a uniform sample can be written as
$$
\frac{\partial{T}}{\partial{t}}= \frac{k}{c_p \rho}\frac{\partial^2{T}}{\partial{x}^2}
$$
where $c_p$ is the constant pressure specific heat and $\rho$ the mass density.
This is a partial differential equations and methods are known to solve it.
However, from your question I am wondering if you have in mind a picture like this (coming from youtube:
In the context of Fourier's law, the two "plates" have not to be intended as two physical objects, but are just two geometric surfaces of the same material which fills the whole space in between, used to evaluate the integral of the heat flux density.
The physical problem of finding the temperature field between two $physical$ plates at different temperature, one in front of the other, may require more than the Fourier law (for example, if between the two surfaces there would be a gas, convective motion could play an important role).