Let $A$ be a $k$-algebra, for instance $A=k[x]$. Then we know that a module $M$ is indecomposable iff $\operatorname{End}_A(M)$ is a local ring (whose maximal ideal is denoted by $\mathfrak m$). This implies that $E:=\operatorname{End}_A(M)/\mathfrak m$ is a $k$-division algebra, which might be strictly larger than $k$. For instance, let $k=\mathbf F_2, A=k[x]$ and $M=A/(x^2+x+1)$ such that $E=\mathbf F_4$.
Now, let $A$ be a graded $k$-algebra, and consider graded $A$-modules. Ideally, $A$ could be some $k[x_1,\dotsc,x_n]$ with an $\mathbf N^n$-grading.
Question Are there indecomposable graded $A$-modules for which $E$ is strictly larger than $k$?
For $n=1$, I won't find something like this, since all indecomposable f.g. $k[x]$-modules are of the form $(x^n)/(x^{n+m})$ with endomorphism ring isomorphic to $k$.