For $n$ an integer greater than $2$, Can one always get a complete theory over a finite language with exactly $n$ models (up to isomorphism)?

There’s a theorem that says that $2$ is impossible.

My understanding is this should be doable in a finite language, but I don’t know how.

If you switch to a countable language, then you can do it as follows. To get $3$ models, take the theory of unbounded dense linear orderings together with a sequence of increasing constants $\langle c_i: i < \omega\rangle$. Then the $c_i$’s can either have no upper bound, an upper bound but no sup, or have a sup. This gives exactly $3$ models. To get a number bigger than $3$, we include a way to color all elements, and require that each color is unbounded and dense. (The $c_i$’s can be whatever color you like.) Then, we get one model for each color of the sup plus the two sup-less models.