Can I do differential gene expression analysis by using the cases and control data from two different GSEs?
If what you are asking is whether you can compare control and treatment across different experiments then the answer is that you have to more careful (while you are doing it) and more transparent (when disseminating the results) about what you are doing.

The statistical power of most methods will be greatly reduced and you would need to perform more careful error analysis and estimation.

What you are asking about is a variant of meta-analysis, where one combines the results of multiple, prior scientific studies to derive new insights.