Turing machine to output enumeration of a language
Here is one way to approach this problem. Program a machine that acts as follows:


Initialize the tape with $\#01$ (or $\epsilon\#01$).
At each step, write $\#$, then go back to the previous $\#$ and copy the string written there, say $0^n1^n$.
Change the first $1$ to a $0$, add two $1$s at the end, and go to the next step.